RodabaughAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 1, 2008

pp. 77-108

Functorial comparisons of bitopology with
topology and the case for redundancy of
bitopology in lattice-valued mathematics

S. E. Rodabaugh
∗

Abstract. This paper studies various functors between (lattice-
valued) topology and (lattice-valued) bitopology, including the ex-
pected “doubling” functor Ed : L-Top → L-BiTop and the “cross”
functor E× : L-BiTop → L

2-Top introduced in this paper, both
of which are extremely well-behaved strict, concrete, full embeddings.
Given the greater simplicity of lattice-valued topology vis-a-vis lattice-
valued bitopology and the fact that the class of L2-Top’s is strictly
smaller than the class of L-Top’s encompassing fixed-basis topology,
the class of E×’s makes the case that lattice-valued bitopology is cat-
egorically redundant. As a special application, traditional bitopology
as represented by BiTop is (isomorphic in an extremely well-behaved
way to) a strict subcategory of 4-Top, where 4 is the four element
Boolean algebra; this makes the case that traditional bitopology is a
special case of a much simpler fixed-basis topology.

2000 AMS Classification: Primary: 03E72, 54A10, 54A40, 54E55.
Secondary: 06F07, 18A22, 18A30, 18A35, 18B30.

Keywords: unital-semi-quantale, unital quantale, (fixed-basis) topology,
(fixed-basis) bitopology, order-isomorphism, categorical (functorial) embed-
ding, redundancy.

1. Introduction and preliminaries

1.1. Motivation. Bitopology has a long and distinguished history spanning
five decades and a literature of some 700 papers [29] with traditional bitopology
playing a wide range of roles in Baire spaces, homotopy and algebraic topology,
generalizations of metric spaces, biframes, programming semantics, etc.

∗Support of Youngstown State University via a sabbatical for the 2005–2006 academic
year is gratefully acknowledged.



78 S. E. Rodabaugh

First defined and used in [31, 32, 3, 4], a bitopological space was originally
defined as a triple ((X, T) , (X, S) , e) with (X, T) , (X, S) topological spaces
and e : (X, T) → (X, S) a continuous bijection—cf. [3]. But if we set

T
′ = {e→ (U ) : U ∈ T} ,

then T′ is a topology on X and the continuity of e insures that idX : (X, T
′) →

(X, S) is continuous, i.e., that T′ ⊃ S. It is therefore not surprising that
almost immediately [4] the original definition was replaced by the simpler,
equivalent definition that a bitopological space is a triple (X, T, S) with T, S
topologies on X with T ⊃ S, T being called the strong topology and S the weak
topology. Even in the broader lattice-valued topology setting, this definition
plays a categorical role (Proposition 3.5 below).

Since a quasi-pseudo-metric p on a set X determines its conjugate quasi-
pseudo-metric q, namely by q (x, y) = p (y, x) , quasi-pseudo-metrics necessarily
occur in conjugate pairs which generate pairs of topologies that need not be
related. Thus the definition of a traditional bitopological space was generalized
in [22] to its modern form to be an ordered triple (X, T, S) with T, S topologies
on X (and no relationship assumed between T and S). Further, a bicontinuous
mapping f : (X, T1, T2) → (Y, S1, S2) is a mapping f : X → Y satisfying

T1 ⊃ (f
←)
→

(S1) , T2 ⊃ (f
←)
→

(S2) ,

i.e., f : (X, T1) → (Y, S1) and f : (X, T2) → (Y, S2) are both continuous. With
the composition and identities of Set, one has the category BiTop, which is
a topological construct and hence strongly complete and strongly cocomplete
along with many other properties.

There is a voluminous literature for BiTop concerning separation, com-
pactness, connectedness, completion, connections to uniform and quasi-uniform
spaces, homotopy groups and algebraic topology, relationships to bilocales [2],
a recently emerging role in programming semantics [25], etc. A significant
part of the recent literature on bitopology is in lattice-valued mathematics
[30, 27, 50, 51]. Letting L be a us-quantale (Subsection 1.2 below) and X a
set, the triple (X, τ, σ) is an L-bitopological space if τ, σ are L-topologies on X
(Subsection 1.5); and such spaces with L-bicontinuous mappings comprise the
category L-BiTop. This category is a topological construct, strongly complete,
strongly cocomplete, and so on. The schemum {L-BiTop : L ∈ |USQuant|}
essentially includes BiTop via its functorial isomorph 2-BiTop.

This paper studies functorial relationships between (lattice-valued) bitopol-
ogy and (lattice-valued) topology in Sections 2–3. The expected functor Ed
strictly embeds L-Top into L-BiTop, a functor we dub the “doubling” func-
tor; and to fully study Ed, it is necessary to construct several functors from
L-BiTop to L-Top whose relationships with Ed lead us to conclude that Ed
is extremely well-behaved. But on the other hand, for each L ∈ |USQuant| ,
the direct product L2 ∈ |USQuant| and there is an embedding E× of L-
BiTop into L2-Top (3.4.1) which is extremely well-behaved (Subsubsections
3.4.2, 3.4.3) if L is a u-quantale (Subsection 1.2) and a strict embedding if L



Bitopology as topology 79

is consistent (Subsubsection 3.4.1). Given that this embedding is strict (for
consistent L) and that the L2’s form a proper subclass of USQuant—which
means (lattice-valued) bitopology is properly “contained” in the proper sub-
class

{

L2-BiTop : L ∈ |USQuant|
}

of {L-Top : L ∈ |USQuant|} , it follows
(lattice-valued) topology (twice) strictly generalizes bitopology. In Section 4
we summarize some metamathematical facts: given that lattice-valued topol-
ogy is fundamentally simpler than lattice-valued bitopology—a membership
lattice and one topology vis-a-vis a membership lattice and two topologies,
it follows that topology and the class of embeddings E×’s make lattice-valued
bitopology categorically redundant; and as a special application, traditional
bitopology BiTop strictly embeds in an extremely well behaved way into 4-
Top, the latter being lattice-valued topology based on the four-element Boolean
algebra 4, so that traditional bitopology both is a strictly special case of the
simpler lattice-valued topology and demonstrates the necessity of lattice-valued
topology. On the other hand, this last fact points the way for bringing over
into lattice-valued topology successful ideas from the extensive literature of
traditional bitopology; in particular, traditional bicompactness mandates, via
the embedding of BiTop into 4-Top, the compactness of [5] for lattice-valued
topology (Corollary 4.7).

1.2. Lattice theoretics. A semi-quantale (L,≤,⊗) (s-quantale) is a com-
plete lattice (L,≤) equipped with a binary operation ⊗ : L × L → L, with no
additional assumptions, called a tensor product; an ordered semi-quantale
(os-quantale) is an s-quantale in which ⊗ is isotone in both variables; a com-
plete quasi-monoidal lattice (cqml) [20, 41] is an os-quantale for which ⊤
is an idempotent element for ⊗; a unital semi-quantale (us-quantale) is an
s-quantale in which ⊗ has an identity element e ∈ L called the unit [33]—units
are unique; a quantale is an s-quantale with ⊗ associative and distributing
across arbitrary

∨

from both sides (implying ⊥ is a two-sided zero) [20, 33, 49];
and a unital quantale (u-quantale) is a us-quantale which is a quantale; and
a strictly two-sided quantale (st-quantale) is a u-quantale for which e = ⊤
[20]. All quantales are os-quantales. The notions of s-quantales, os-quantales,
and us-quantales are from [45, 46].

SQuant comprises all semi-quantales together with mappings preserving
⊗ and arbitrary

∨

; OSQuant is the full subcategory of SQuant of all os-
quantales; USQuant is a subcategory of SQuant comprising all us-quantales
together with all mappings preserving arbitrary

∨

, ⊗, and e; Quant is the full
subcategory of OSQuant of all quantales; and UQuant is the full subcategory
of UOSQuant of all unital quantales. Note uos-quantales for which ⊗ = ∧
(binary) are semiframes and SFrm is the full subcategory of UOSQuant
of all semiframes; and u-quantales for which ⊗ = ∧ (binary) are frames—
in which case e = ⊤—and Frm is the full subcategory of UQuant of all
frames. Semiframes equipped with an order-reversing involution are com-
plete DeMorgan algebras; and s-quantales equipped with a semi-polarity



80 S. E. Rodabaugh

[16] (∀α, β ∈ L, α ≤ β ⇒ β′ ≤ α′ and α ≤ (α′)
′
) are complete semi-

DeMorgan s-quantales.
Throughout this paper, the requirement of us-quantale [u-quantale] can be

relaxed to s-quantale [quantale, resp.] if one wishes to consider the relationships
between q-topology and q-bitopology ([46] and Subsection 1.5 below).

Justifying the above lattice-theoretic notions is a wealth of examples (see
[17, 20, 21, 23, 33, 35, 39, 40, 41, 44, 46] and their references). The lattice
2 = {⊥,⊤} with ⊥ 6= ⊤; and a lattice is consistent if it contains 2 and
inconsistent if it is singleton (with ⊥ = ⊤).

1.3. Powerset operators. Let X ∈ |Set| and L ∈ |SQuant|. Then LX is the
L-powerset of X of all L-subsets of X. The constant L-subset member of
LX having value α is denoted α. All order-theoretic operations (e.g.,

∨

,
∧

) and
algebraic operations (e.g., ⊗) on L lift point-wise to LX and are denoted by
the same symbols. In the case L ∈ |USQuant| , the unit e lifts to the constant
map e, which is the unit of ⊗ as lifted to LX .

The operator ℘∅ : |Set| → |Set| is useful in this paper, where ℘∅ (X)
denotes the poset of all the nonempty subsets of X.

Let L ∈ |SQuant| , X, Y ∈ |Set| , and f : X → Y be in Set. Then the
standard (traditional) image and preimage operators f→ : ℘ (X) → ℘ (Y ) ,
f← : ℘ (X) ← ℘ (Y ) are

f→ (A) = {f (x) ∈ Y : x ∈ A} , f← (B) = {x ∈ X : f (x) ∈ B} ,

and the Zadeh image and preimage operators f→L : L
X → LY , f←L : L

X ← LY

[53] are

f→L (a) (y) =
∨

{a (x) : x ∈ f← ({y})} , f←L (b) = b ◦ f.

If L is understood, it may be dropped providing the context distinguishes these
operators from the traditional operators. It is observed that f→ and f← are
naturally isomorphic to f→2 and f

←
2 , resp.

It is well-known [36, 37, 39, 40, 46] that each f←L preserves arbitrary
∨

,
arbitrary

∧

, ⊗, and all constant maps, as well as the unit e if L ∈ |USQuant|;
each f→L preserves arbitrary

∨

;

f→ ⊣ f←, f→L ⊣ f
←
L ;

f→ and f→L are left-inverses [right-inverses] of f
← and f←L , resp., if f is sur-

jective [injective, resp.]; and f→, f←, f→L , f
←
L are all order-isomorphisms if and

only if f is a bijection.
Powerset operators and the powerset theories underlying lattice-valued math-

ematics are studied extensively in [6, 14, 7, 8, 15, 10, 11, 36, 37, 39, 40, 46].

1.4. Category theoretics. The main reference for categorical notions is [1],
to which we refer the reader for various properties of functors as well as various
versions of the Adjoint Functor Theorem and related notions.

The proving of functorial adjunctions is done via lifting (or major) and
naturality (or minor) diagrams in the manner of [28, 36, 37, 41].



Bitopology as topology 81

1.5. Topology and bitopology. Given L ∈ |USQuant|, the category L-Top
comprises objects of the form (X, τ ), where τ ⊂ LX is closed under arbitrary
∨

and binary ⊗ and contains e—so that τ is a sub-us-quantale of LX , together
with morphisms f : (X, τ ) → (Y, σ), where f : X → Y is a function and

τ ⊃ (f←L )
→

(σ) ,

namely f←L (v) ∈ τ for each v ∈ σ. The objects (X, τ ) are called L-topological
spaces and τ is an L-topology on X comprising L-subsets of X; and the
morphisms f are called L-continuous. Cf. [20, 41, 46].

Similarly, the category L-BiTop comprises objects of the form (X, τ, σ) ,
where τ, σ are L-topologies on X, together with morphisms f : (X, τ1, τ2) →
(Y, σ1, σ2), where f : X → Y is a function and

τ1 ⊃ (f
←
L )
→

(σ1) , τ2 ⊃ (f
←
L )
→

(σ2) .

The objects (X, τ, σ) are called L-bitopological spaces and (τ, σ) is an L-
bitopology on X; and the morphisms f are called L-bicontinuous. If the L
is clear in context, it may be dropped from the labels.

As noted in Subsection 1.1, the traditional category BiTop is isomorphic to
2-BiTop (cf. 3.25 below) and embeds into each L-BiTop, and similarly Top
is isomorphic to 2-Top and embeds into each L-Top.

Each of L-Top and L-BiTop has the base L of the category fixed and so
is part of fixed-basis (lattice-valued) topology and fixed-basis (lattice-valued)
bitopology, resp. The disciplines of fixed-basis topology and fixed-basis bitopol-
ogy are encompassed by the respective classes

{L-Top : L ∈ |USQuant|} , {L-BiTop : L ∈ |USQuant|} .

Both L-Top and L-BiTop are topological over Set and have small fi-
bres, hence are [co]complete and [co]well-powered, and hence are strongly
[co]complete with many other nice properties (see 3.36 and its proof below).
The categorical product for L-Top is given in, or adapted from, [12, 52] (cf.
[20, 41]) and for {(Xγ , τγ ) : γ ∈ Γ} denoted by





∏

γ ∈Γ

(Xγ , τγ ) ,{πγ}γ ∈Γ



 ,
∏

γ ∈Γ

(Xγ , τγ ) ≡ (×γ ∈ΓXγ , Πγ ∈Γτγ ) ,

where {πγ : γ ∈ Γ} are the projections. The binary L-topological product for
two spaces (X, τ ) , (Y, σ) is denoted (X, τ ) Π (Y, σ) or (X × Y, τ Π σ) with pro-
jections {π1, π2}. The categorical product for L-BiTop for {(Xγ , τγ ) : γ ∈ Γ}
is





∏

γ ∈Γ

(Xγ , τγ , σγ ) ,{πγ}γ ∈Γ



 ,

∏

γ ∈Γ

(Xγ , τγ , σγ ) ≡ (×γ ∈ΓXγ , Πγ ∈Γτγ , Πγ ∈Γσγ ) ,

where Πγ ∈Γτγ , Πγ ∈Γσγ are the L-topological product topologies in each slot
and the projections are as above.



82 S. E. Rodabaugh

An L-topology τ is weakly stratified [20] if {α : α ∈ L} ⊂ τ , non-
stratified if it is not weakly stratified, and anti-stratified [9, 35] if

{α : α ∈ L, α ∈ τ} = {⊥, e} ;

so a weakly stratified topology contains all constant L-subsets, while an anti-
stratified topology contains precisely the constant L-subsets ⊥ and e (which
are the same if L is inconsistent with ⊥ = ⊤). An L-topological space is weakly
stratified [anti-stratified] if its topology is weakly stratified [anti-stratified], and
an L-bitopological space is weakly stratified [anti-stratified] if both topologies
are weakly stratified [anti-stratified]. The inclusionist position that the axioms
of a fixed-basis topology must allow for all types of stratification has recently
received additional, emphatic confirmations from both lattice-valued frames
[35] and topological systems in domain theory [9].

The following definition and proposition are needed in this paper.

Definition 1.1. Let X be a set and let L be a us-quantale. Then the L-
topological fibre, respectively, L-bitopological fibre on X is

L-T (X) ≡
{

τ ⊂ LX : (X, τ ) ∈ |L-Top|
}

,

L-BT (X) ≡ {(τ, σ) : (X, τ, σ) ∈ |L-BiTop|} .

Proposition 1.2. Let X be a set, let L be a us-quantale, and recall ℘∅ from
Subsection 1.3.

(1) L-T (X) is a complete meet subsemilattice of ℘
(

LX
)

; and since each

L-topology is nonempty, L-T (X) ⊂ ℘∅
(

LX
)

.
(2) L-BT (X), ordered coordinate-wise by inclusion, is a complete meet

subsemilattice of ℘
(

LX
)

×℘
(

LX
)

; and further, L-BT (X) ⊂ ℘∅
(

LX
)

×

℘∅
(

LX
)

.

Proof. The first part of (1) is well-known, and the second part of (1) is trivial.
Now (2) follows from (1) since

L-BT (X) = L-T (X) × L-T (X) ⊂ ℘∅
(

LX
)

× ℘∅
(

LX
)

.

�

Finally, we need the notion of a subbase of an L-topology τ on X [41]. We
say σ ⊂ LX is a subbase of τ , written τ = 〈〈σ〉〉 , if

τ =
⋂

{τ′ ∈ L-T (X) : σ ⊂ τ′} ,

the right-hand side always existing by Proposition 1.2(1), and we say β ⊂ LX

is a base of τ , written τ = 〈β〉 , if

∀u ∈ τ, ∃Bu ⊂ β, u =
∨

Bu.

One can always pass from a subbase σ to a topology τ through a base β in the
traditional way, written

τ = 〈β〉 = 〈〈σ〉〉 ,

if and only if ⊗ is associative and distributes across arbitrary
∨

, i.e., if and
only if L is a u-quantale.



Bitopology as topology 83

2. Functorial interpretations of topology as bitopology

For each us-quantale L, this section records a simple (and expected) “dou-
bling” embedding Ed : L-Top → L-BiTop. The behavior of Ed w.r.t. limits
and colimits—it preserves, reflects, detects both—is examined completely in
Subsections 3.1–3.2 below. It emerges that Ed is an extremely well-behaved
embedding.

Proposition 2.1. Let L be a us-quantale. Define Ed : L-Top → L-BiTop by
the following correspondences:

Ed (X, τ ) = (X, τ, τ ) , Ed (f ) = f.

Then Ed is a concrete, full, strict embedding; and so L-Top is isomorphic to
a full subcategory of L-BiTop.

Proof. All details are straightforward. �

3. Functorial interpretations of bitopology as topology

This section records several interpretations of bitopology as topology, the
most important of which would seem to be the extremely well-behaved embed-
ding E× of Subsection 3.4.

3.1. Fl, Fr, F∧ : L-BiTop → L-Top and behavior of Ed : L-Top → L-
BiTop w.r.t. limits. This subsection constructs the concrete, faithful, full
forgetful functors—the “left-forgetful” functor Fl : L-BiTop → L-Top and
the “right-forgetful” Fr : L-BiTop → L-Top—as well as the concrete, faithful
“meet” functor F∧ : L-BiTop → L-Top and shows F∧ is the left-adjoint of
Ed of the previous section and that each of Fl, Fr is a left-adjoint of Ed under
certain restrictions.

Proposition 3.1. Let L be a us-quantale and define Fl, Fr : L-BiTop → L-
Top as follows:

Fl (X, τ, σ) = (X, τ ) , Fl (f ) = f,

Fr (X, τ, σ) = (X, σ) , Fr (f ) = f.

Then each of Fl, Fr is a concrete, faithful, full, object-surjective functor, but
need not be an embedding.

Proof. We comment only on Fl. Trivially, Fl is a concrete, faithful, object-
surjective functor. As for fullness, let f : (X, τ ) → (Y, σ) in L-Top; then
f : (X, τ, τ ) → (Y, σ, σ) is L-bicontinuous, so is in L-BiTop, and maps to
f : (X, τ ) → (Y, σ). Now suppose that either [|X| ≥ 1 and |L| ≥ 3] or [|X| ≥ 2
and |L| ≥ 2]; then ∃τ, σ ∈ L-T (X) with τ 6= σ, so that Fl (X, τ, σ) = (X, τ ) =
Fl (X, τ, τ ), and hence Fl does not inject objects and is not an embedding. �

Proposition 3.2. Let L be a us-quantale and define F∧ : L-BiTop → L-Top
as follows:

F∧ (X, τ, σ) = (X, τ ∩ σ) , F∧ (f ) = f.



84 S. E. Rodabaugh

Then F∧ is a concrete, faithful, object-surjective functor, but need not be full
nor an embedding.

Proof. Since L-Top has complete fibres, F∧ is well-defined on objects. Now let
f : (X, τ1, τ2) → (Y, σ1, σ2) be L-bicontinuous. Since the image operator of the
Zadeh preimage operator preserves ⊂, it follows

τ1 ⊃ (f
←
L )
→

(σ1) , τ2 ⊃ (f
←
L )
→

(σ2) ⇒

τ1 ∩ τ2 ⊃ (f
←
L )
→

(σ1) ∩ (f
←
L )
→

(σ2) ⊃ (f
←
L )
→

(σ1 ∩ σ2) ,

so that f : (X, τ1 ∩ τ2) → (Y, σ1∩σ2) is L-continuous. Immediately, F∧ is a
concrete, faithful functor which surjects objects.

Now let L be a complete DeMorgan algebra (with ⊗ = ∧ (binary)) and
consider each of the L-bitopological spaces (R (L) , τl (L) , τl (L)) and
(R (L) , τl (L) , τ (L)) , where R (L) is the L-fuzzy real line, τl (L) is the left-hand
L-topology on R (L) determined by the Lt operators and τ (L) is the standard
L-topology on R (L) [43]. Then (R (L) , τl (L) , τl (L)) 6= (R (L) , τl (L) , τ (L))
and

F∧ (R (L) , τl (L) , τl (L)) = (R (L) , τl (L))

= (R (L) , τl (L) ∩ τ (L))

= F∧ (R (L) , τl (L) , τ (L)) ,

showing that F∧ does not inject objects, so is not an embedding. Now letting
f : R (L) → R (L) be idR(L), we have

f : F∧ (R (L) , τl (L) , τ (L)) → F∧ (R (L) , τ (L) , τl (L))

is L-continuous, but

f : (R (L) , τl (L) , τ (L)) → (R (L) , τ (L) , τl (L))

cannot be L-bicontinuous (because of the first slot). The concreteness of F∧
implies there exists no g ∈ L-BiTop with F∧ (g) = f , so F∧ is not full. �

Theorem 3.3. Let L be a us-quantale. Then F∧ ⊣ Ed, this adjunction is a
monocoreflection, and F∧ takes L-BiTop to a monocoreflective subcategory of
L-Top. On the other hand, Ed ⊣/ F∧.

Proof. Let (X, τ1, τ2) ∈ |L-BiTop| , choose

η = id : (X, τ1, τ2) → EdF∧ (X, τ1, τ2) = (X, τ1 ∩ τ2, τ1 ∩ τ2) ,

and note η is an L-continuous injection. Now let (Y, σ) ∈ |L-Top| , suppose
f : (X, τ1, τ2) → Ed (Y, σ) = (Y, σ, σ) is L-bicontinuous, and note

τ1 ⊃ (f
←
L )
→

(σ) , τ2 ⊃ (f
←
L )
→

(σ) ⇒ τ1 ∩ τ2 ⊃ (f
←
L )
→

(σ) ,

making f : F∧ (X, τ1, τ2) = (X, τ1 ∩ τ2) → (Y, σ) L-continuous. Then f = f
is the unique choice making f = f ◦ η. The naturality diagram now follows
by concreteness as do the other claims concerning F∧ ⊣ Ed. Finally, given F∨
of Subsection 3.2 and Ed ⊣ F∨ of 3.9 below, Ed ⊣/ F∧ since F∧ ≇ F∨ and
right-adjoints are essentially unique. �



Bitopology as topology 85

Definition 3.4. L-BiTop (⊂) [L-BiTop (⊃)] is the full subcategory of L-
BiTop of all spaces (X, τ, σ) in which τ ⊂ σ [τ ⊃ σ].

Note BiTop (⊂) and BiTop (⊃) (essentially setting L = 2) express the
original sense of traditional bitopology [3, 4].

Proposition 3.5. Let L be a us-quantale. Then

Fl |L-BiTop(⊂) = F∧|L-BiTop(⊂), Fr |L-BiTop(⊃) = F∧|L-BiTop(⊃).

Hence Fl |L-BiTop(⊂) ⊣ Ed and Fr |L-BiTop(⊃) ⊣ Ed, but Ed ⊣/ Fl |L-BiTop(⊂)
and Ed ⊣/ Fr |L-BiTop(⊃).

Proof. The restricted forgetful functors obviously coincide with the meet func-
tor. Observing that Ed maps into each of L-BiTop (⊂) and L-BiTop (⊃), the
claimed adjunctions are then immediate from 3.3. The claimed non-adjunctions
follow from 3.10 below. �

Corollary 3.6. Let L be a us-quantale. The following hold:

(1) Ed preserves all strong limits and F∧ preserves all strong colimits.
(2) Fl preserves the strong colimits of L-BiTop (⊂), Fr preserves the strong

colimits of L-BiTop (⊃), and Ed preserves strong limits into each of
L-BiTop (⊂) and L-BiTop (⊃).

Proposition 3.7. For each us-quantale L, Ed : L-Top → L-BiTop reflects
and detects all limits and hence lifts all limits and is transportable.

Proof. The details are straightforward using 3.6 and Proposition 13.34 [1]. �

3.2. F∨ : L-BiTop → L-Top and behavior of Ed : L-Top → L-BiTop
w.r.t. colimits. This subsection constructs the concrete, faithful “join” func-
tor F∨ : L-BiTop → L-Top and shows it is the right-adjoint of Ed of the
previous section.

Proposition 3.8. Let L be a us-quantale and define F∨ : L-BiTop → L-Top
as follows:

F∨ (X, τ, σ) = (X, τ ∨ σ) , F∨ (f ) = f,

where

τ ∨ σ = 〈〈τ ∪ σ〉〉

Then F∨ is a concrete, faithful, object-surjective functor, but need not be full
nor an embedding.

Proof. Since L-Top has complete fibres, F∨ is well-defined on objects. Now let
f : (X, τ1, τ2) → (Y, σ1, σ2) be L-bicontinuous. Since the image operator of the
Zadeh preimage operator preserves unions, then

τ1 ⊃ (f
←
L )
→

(σ1) , τ2 ⊃ (f
←
L )
→

(σ2) ⇒

τ1 ∨ τ2 ⊃ τ1 ∪ τ2 ⊃ (f
←
L )
→

(σ1) ∪ (f
←
L )
→

(σ2) = (f
←
L )
→

(σ1 ∪ σ2) ,



86 S. E. Rodabaugh

so that f : (X, τ1 ∨ τ2) → (Y, σ1 ∨ σ2) is L-subbasic continuous. By Theorem
3.2.6 of [41] as restricted to the fixed-basis case and then adapted to the us-
quantalic case, f : (X, τ1 ∨ τ2) → (Y, σ1 ∨ σ2) is L-continuous. Immediately,
F∨ is a concrete, faithful functor which surjects objects.

Now let L be a complete DeMorgan algebra (with ⊗ = ∧ (binary)) and con-
sider each of the L-bitopological spaces (R (L) , τl (L) , τr (L)) and
(R (L) , τ (L) , τ (L)) , where R (L) is the L-fuzzy real line, τl (L) is the left-hand
L-topology on R (L) determined by the Lt operators, τr (L) is the left-hand L-
topology on R (L) determined by the Rt operators, and τ (L) is the standard
L-topology on R (L) [43]. Then (R (L) , τl (L) , τr (L)) 6= (R (L) , τ (L) , τ (L))
and

F∨ (R (L) , τl (L) , τr (L)) = (R (L) , τl (L) ∨ τr (L))

= (R (L) , τ (L))

= F∨ (R (L) , τ (L) , τ (L)) ,

showing that F∨ does not inject objects, so is not an embedding. Now letting
f : R (L) → R (L) be idR(L), we have

f : F∨ (R (L) , τl (L) , τr (L)) → F∨ (R (L) , τr (L) , τl (L))

is L-continuous, but

f : (R (L) , τl (L) , τr (L)) → (R (L) , τr (L) , τl (L))

cannot be L-bicontinuous. The concreteness of F∨ implies there exists no
g ∈ L-BiTop with F∨ (g) = f , so F∨ is not full. �

Theorem 3.9. Let L be a us-quantale. Then Ed ⊣ F∨, this adjunction is an
isoreflection, and F∨ takes L-BiTop to an isoreflective subcategory of L-Top.
On the other hand, F∨ ⊣/ Ed.

Proof. Let (X, τ ) ∈ |L-Top| , choose

η = id : (X, τ ) → F∨Ed (X, τ ) = (X, τ ∨ τ ) = (X, τ ) ,

and note η is an L-homeomorphism. Now let (Y, σ1, σ2) ∈ |L-BiTop| , suppose
f : (X, τ ) → F∨ (Y, σ1, σ2) = (Y, σ1 ∨ σ2) is L-continuous, and note

τ ⊃ (f←L )
→

(σ1 ∨ σ2) ⊃ (f
←
L )
→

(σ1 ∪ σ2) ⊃ (f
←
L )
→

(σ1) , (f
←
L )
→

(σ2) ,

making f : Ed (X, τ ) = (X, τ, τ ) → (Y, σ1, σ2) L-bicontinuous. Then f = f
is the unique choice making f = f ◦ η. The naturality diagram now follows
by concreteness, as do the other claims concerning Ed ⊣ F∨. Finally, given
F∧ of Subsection 3.1 and F∧ ⊣ Ed of 3.3 above, F∨ ⊣/ Ed since F∧ ≇ F∨ and
left-adjoints are essentially unique. �

Corollary 3.10. Let L be a us-quantale. The following hold:

(1) Ed preserves all strong colimits and F∨ preserves all strong limits.
(2) Ed ⊣/ Fl |L-BiTop(⊂) and Ed ⊣/ Fr |L-BiTop(⊃), and hence Ed ⊣/ Fl and

Ed ⊣/ Fr.



Bitopology as topology 87

Proof. (1) is immediate. As for (2), it is clear that Fl |L-BiTop(⊂), Fr |L-BiTop(⊃)
≇ F∨ |L-BiTop(⊂), F∨ |L-BiTop(⊃), resp., implying Ed ⊣/ Fl |L-BiTop(⊂) and
Ed ⊣/ Fr |L-BiTop(⊃) by the essential uniqueness of the right-adjoint in 3.9; and
hence Ed ⊣/ Fl and Ed ⊣/ Fr. �

Proposition 3.11. For each us-quantale L, Ed : L-Top → L-BiTop reflects
and detects all colimits.

Proof. The details are straightforward. �

3.3. FΠ : L-BiTop → L-Top. This subsection constructs the non-concrete,
faithful “product” functor FΠ : L-BiTop → L-Top which, when appropriately
restricted, is an embedding. It need not preserve finite products and hence
lacks a left-adjoint.

Proposition 3.12. Let L be a us-quantale and define FΠ : L-BiTop → L-Top
as follows:

FΠ (X, τ, σ) = (X × X, τ Π σ) , FΠ (f ) = f × f,

where τ Π σ is the L-product topology on X × X (Subsection 1.5). Then FΠ is
a non-concrete, faithful functor which need not be full nor object-surjective nor
an embedding.

Proof. Immediately FΠ is well-defined on objects. Let f : (X, τ1, τ2) →
(Y, σ1, σ2) be L-bicontinuous and let v ∈ σ1 Π σ2 be a subbasic open set of
the form (π1)

←
L

(s1) with s1 ∈ σ1. Then given (x1, x2) ∈ X × X,

(f × f )
←
L

(v) (x1, x2) = (π1)
←
L

(s1) (f (x1) , f (x2))

= s1 (π1 (f (x1) , f (x2)))

= s1 (f (x1))

= f←L (s1) (x1)

= f←L (s1) (π1 (x1, x2))

= (π1)
←
L

(f←L (s1)) (x1, x2) ,

so that (f × f )
←
L

(v) = (π1)
←
L

(f←L (s1)) ∈ τ1 Π τ2; and similarly, if v is a sub-
basic open set of the form (π2)

←
L

(s2) with s2 ∈ σ2, (f × f )
←
L

(v) ∈ τ1 Π τ2.
So FΠ (f ) : FΠ (X, τ1, τ2) → FΠ (Y, σ1, σ2) is L-subbasic continuous and hence
L-continuous (cf. Theorem 3.2.6 of [41]). It is easy to show FΠ preserves com-
position and identities—and so is a functor—and is faithful and need not be
full nor object-surjective.

To see that FΠ need not inject objects, let L = {⊥, α, β,⊤} be a chain
with ⊗ = ∧ (binary), X = {x} , τ1 = {⊥, α,⊤} , and τ2 =

{

⊥, β,⊤
}

. Then
(X, τ1, τ2) 6= (X, τ2, τ1) , yet FΠ (X, τ1, τ2) = FΠ (X, τ2, τ1). �

Proposition 3.13. FΠ does not preserve binary products and hence has no
left-adjoint.

Proof. Let (X, τ1, τ2) , (Y, σ1, σ2) be given with X 6= Y . Then the carrier set
of FΠ [(X, τ1, τ2) Π (Y, σ1, σ2)] is (X × Y ) × (X × Y ) and the carrier set of
FΠ (X, τ1, τ2) Π FΠ (Y, σ1, σ2) is (X × X)×(Y × Y ), clearly not the same. �



88 S. E. Rodabaugh

Definition 3.14. Letting L be a us-quantale, L-NBiTop is the full subcategory
of all spaces (X, τ, σ) satisfying the condition that each open L-subset u 6= ⊥
in each of τ, σ is L-normalized, i.e., has the property that

∨

x∈X

u (x) = e.

If L is an st-quantale, then the notion of L-normalized subsets coincides with
the usual notion, namely

∨

x∈X u (x) = ⊤.

Theorem 3.15. Let L be a u-quantale. Then FΠ |L-NBiTop : L-NBiTop → L-
Top is an embedding. This embedding does not preserve binary products and
hence has no left-adjoint.

Proof. Because of 3.12, it suffices to show FΠ as restricted injects objects. For
two distinct objects, let us consider

(X, τ1, σ) 6= (X, τ2, σ)

with τ1 6= τ2; all other cases are similar and left to the reader. Suppose
W.L.O.G. there is u ∈ τ1 − τ2 and assume τ1 Π σ = τ2 Π σ on X × X. Then
setting ⊠ ≡⊗◦×,

∃ {uγ ⊠ vγ}γ ∈Γ ⊂ τ2 Π σ

such that
(π1)

←
L

(u) =
∨

γ ∈Γ

(uγ ⊠ vγ ) .

Applying the surjectivity of π1 and properties of Zadeh image operators (Sub-
section 1.3), we obtain the contradiction

u = (π1)
→
L

((π1)
←
L

(u))

= (π1)
→
L





∨

γ ∈Γ

(uγ ⊠ vγ )





=
∨

γ ∈Γ

((π1)
→
L

(uγ ⊠ vγ ))

=
∨

γ ∈Γ

uγ ∈ τ2,

where we have used the fact, for each γ ∈ Γ and each x ∈ X, that

(π1)
→
L

(uγ ⊠ vγ ) (x) =
∨

y ∈X

(uγ ⊠ vγ ) (x, y)

=
∨

y ∈X

(uγ (x) ⊗ vγ (y))

= uγ (x) ⊗
∨

y ∈X

vγ (y)

= uγ (x) ⊗ e

= uγ (x) .



Bitopology as topology 89

The non-preservation of products follow for the restricted functor as in the
proof of 3.13. �

Corollary 3.16. FΠ ◦ Gχ : BiTop → Top is an embedding. This embedding
does not preserve binary products and hence has no left-adjoint.

Proof. The first statement is a corollary of 3.15 as follows: given any non-empty
subset A of set X, χA : X → 2 is normalized; 2-NBiTop = 2-BiTop; and
Gχ : BiTop → 2-BiTop is a categorical isomorphism. The non-preservation
of products follows for the composite functor as in the proof of 3.13. �

Remark 3.17. Corollary 3.16 furnishes an embedding of BiTop into Top;
but this is not enough to say that Top may be categorically regarded as a
generalization of BiTop since FΠ ◦ Gχ is not sufficiently well-behaved. This
motivates the search for a better behaved embedding of bitopology into topol-
ogy conducted in the next subsection.

3.4. E× : L-BiTop → L
2-Top. This subsection constructs the concrete, full,

strict “cross” embedding E× : L-BiTop → L
2-Top, establishes its behavior

w.r.t. limits and colimits—for appropriate L, E× preserves both and detects
and reflects the former, and shows that E× is essentially neutral w.r.t. strati-
fication issues. It follows that E× is an extremely well-behaved embedding.

3.4.1. Construction of E× : L-BiTop → L
2-Top.

Proposition 3.18 (cf. [16]). Let X be a set.

(1) For each set L the mapping ϕX : L
X × LX →

(

L2
)X

given by

ϕX (a1, a2) = a1 × a2, i.e., ϕX (a1, a2) (x) = (a1 (x) , a2 (x))

is a bijection with inverse mapping ϕ−1
X

: LX × LX ←
(

L2
)X

given by

ϕ−1
X

(a) = (π1 ◦ a, π2 ◦ a) ,

where π1, π2 are the projections from L
2 to L.

(2) If L is a poset, then ϕX is an order-isomorphism.
(3) If L is a semi-DeMorgan s-quantale, then ϕX preserves semi-complements.
(4) If L is an [u]s-quantale, then ϕX is an [u]s-quantalic isomorphism (i.e.,

ϕX also preserves tensor products [and the unit]).

Proof. The details of (1)−(3) are the same as, or analogous to, those of Lemma
4.4.1 of [16]. The details of (4) are straightforward. �

Corollary 3.19. ϕ→X : ℘
(

LX × LX
)

→ ℘
(

(

L2
)X
)

is an order-isomorphism.

Proof. This is immediate from 3.18(1) using Subsection 1.3. �

Proposition 3.20. Let A, B be nonempty sets. Then ζ : ℘∅ (A) × ℘∅ (B) →
℘∅ (A × B) given by

ζ (C, D) = C × D

is an order-isomorphism onto its image, i.e., an order-embedding.



90 S. E. Rodabaugh

Proof. Clearly ζ is well-defined. As for injectivity, let (C1, D1) 6= (C2, D2).
Then there are several cases, and a typical case is C1 6= C2, D1 = D2. Then
W.L.O.G. there is x ∈ C1 − C2. Since D1 = D2 6= ∅, there is y ∈ D1 = D2.
So (x, y) ∈ (C1 × D1) − (C2 × D2) ; hence ζ (C1, D1) 6= ζ (C2, D2). Since all
orderings in question are coordinate-wise, it follows that both ζ and ζ−1 (on
Im (ζ)) are isotone. �

Proposition 3.21. Let X be a set, L be a us-quantale, and ζ denote any
restriction of the ζ of 3.20.

(1) ζ : ℘∅
(

LX
)

×℘∅
(

LX
)

→ ℘∅
(

LX × LX
)

is an order-isomorphism onto
its image.

(2) ζ : L-BT (X) → ℘∅
(

LX × LX
)

is an order-isomorphism onto its im-
age.

Proof. Conjoin Proposition 1.2 and 3.20. �

Lemma 3.22. Let X be a set and L be a us-quantale, and put
E× : L-BT (X) → L

2-T (X) by

E× = ϕ
→
X ◦ ζ.

Then E× is an order-isomorphism onto its image.

Proof. It must be first verified that E× actually maps into L
2-T (X). Let

(τ1, τ2) ∈ L-BT (X). Then τ1, τ2 are L-topologies on X and hence sub-us-
quantales of LX . It is straightforward to check that as direct products,

ζ (τ1, τ2) = τ1 × τ2 ⊂ L
X × LX

and τ1 × τ2 is a sub-us-quantale of L
X × LX . It follows

E× (τ1, τ2) = ϕ
→
X (ζ (τ1, τ2)) = ϕ

→
X (τ1 × τ2) ⊂ ℘

(

(

L2
)X
)

and that E× (τ1, τ2) is a sub-us-quantale of
(

L2
)X

, namely an L2-topology on

X. Hence E× (τ1, τ2) ∈ L
2-T (X).

The remaining claims concerning E× follow from 3.19 and 3.21. �

Theorem 3.23. Let L be a us-quantale, let

f ∈ L-BiTop ((X, τ1, τ2) , (Y, σ1, σ2)) ,

and put
E× (X, τ1, τ2) = (X, E× (τ1, τ2)) , E× (f ) = f.

Then E× : L-BiTop → L
2-Top is a concrete, full embedding; and hence

L-BiTop is concretely isomorphic to a full subcategory of L2-Top. Further, if
L is consistent, E× is a strict embedding (not a functorial isomorphism).

Proof. It is immediate from 3.22 that E× is well-defined at the object-level into
L2-Top. It must be now checked that E× is well-defined at the morphism-level,
i.e., that f : (X, τ1, τ2) → (Y, σ1, σ2) is L-bicontinuous implies
f : (X, E× (τ1, τ2)) → (Y, E× (σ1, σ2)) is L

2-continuous. To that end, let

v ∈ E× (σ1, σ2) = ϕ
→
Y (σ1 × σ2) .



Bitopology as topology 91

Then ∃ (v1, v2) ∈ σ1 × σ2 with v = ϕY (v1, v2) . Now let x ∈ X. Then

f←L (v) (x) = v (f (x))

= ϕY (v1, v2) (f (x))

= (v1 (f (x)) , v2 (f (x)))

= (f←L (v1) (x) , f
←
L (v2) (x)) .

Since f is L-bicontinuous,

u1 ≡ f
←
L (v1) ∈ τ1, u2 ≡ f

←
L (v2) ∈ τ2;

and so choosing

u = ϕX (u1, u2) ∈ E× (τ1, τ2) ,

we have

f←L (v) = u,

finishing the proof that f is L2-continuous.
Since E× is concrete (with respect to the usual forgetful functors), it is im-

mediate that E× is a functor and that E× injects hom-sets. To verify that E×
is full, we show that f : (X, E× (τ1, τ2)) → (Y, E× (σ1, σ2)) is L

2-continuous im-
plies f : (X, τ1, τ2) → (Y, σ1, σ2) is L-bicontinuous. Let v ∈ σ1 and note ⊥∈ σ2.
Then (v,⊥) ∈ σ1×σ2, so that ϕY (v,⊥) ∈ E× (σ1, σ2). Hence f

←
L (ϕY (v,⊥)) ∈

E× (τ1, τ2) by the L
2-continuity of f . It follows ∃u ∈ E× (τ1, τ2) , and hence

∃ (u1, u2) ∈ τ1 × τ2, such that

f←L (ϕY (v,⊥)) = u = ϕX (u1, u2) .

Now let x ∈ X. Then

(u1 (x) , u2 (x)) = ϕX (u1, u2) (x)

= f←L (ϕY (v,⊥)) (x)

= (v (f (x)) ,⊥)

= (f←L (v) (x) ,⊥) ,

so that u1 (x) = f
←
L (v) (x) . It follows f

←
L (v) = u1 ∈ τ1. Similarly, it can be

shown that if v ∈ σ2, then f
←
L (v) ∈ τ2. Hence f is L-bicontinuous.

For E× to be an embedding, it remains to show that E× injects objects. To
that end let (X, τ1, τ2) 6= (Y, σ1, σ2) . If X 6= Y, we are done. So suppose that
X = Y and that (τ1, τ2) 6= (σ1, σ2). Then immediately by 3.22,

E× (τ1, τ2) 6= E× (σ1, σ2) .

It follows that

E× (X, τ1, τ2) 6= E× (Y, σ1, σ2) .

Finally, the strictness of E×, when L is consistent, follows from 3.24 below.
�

Many more properties of E× are developed in the next three subsections
which show that it is an extremely well-behaved embedding.



92 S. E. Rodabaugh

Counterexample 3.24. If E× were to surject objects, then E× would be a
functorial isomorphism. This however is usually not the case. Let L be any
consistent us-quantale, note L ⊃ 2 = {⊥, e}, and consider the L2-topological
space (X, τ ) with X nonempty and τ the indiscrete L2-topology

τ =
{

(⊥,⊥), (e, e)
}

.

Suppose a space (X, E× (τ1, τ2)) from the image of E× is (X, τ ). This forces

ϕ→X (τ1 × τ2) = E× (τ1, τ2) = τ.

Noting

{⊥, e}⊂ τ1, {⊥, e}⊂ τ2,

it follows

ϕX (⊥, e) ∈ ϕ
→
X (τ1 × τ2) , ϕX (⊥, e) = (⊥, e) /∈ τ,

a contradiction. Hence the space (X, τ ) is not in the image of E×. Hence, for
consistent L it is the case that E× is not a functorial isomorphism, but a strict
embedding. This justifies examining E×’s behavior w.r.t. limits and colimits in
the next Subsubsections 3.4.2–3.4.3 as well as characterizing

∣

∣E→× (L-BiTop)
∣

∣

in 3.28.

Corollary 3.25. Let 4 be the 4-element Boolean algebra {⊥, α, β,⊤}with
⊗ = ∧ (binary). Then the traditional category BiTop of bitopological spaces
and bicontinuous maps concretely, fully, strictly embeds into 4-Top as a full
monocoreflective subcategory that is closed under all limits and colimits.

Proof. Consider the bitopological version Gχ : BiTop → 2-BiTop of the cha-
racteristic functor given by

Gχ (T) = {χU : U ∈ T} , Gχ (S) = {χV : V ∈ S} ,

Gχ (X, T, S) = (X, Gχ (T) , Gχ (S)) , Gχ (f ) = f.

Then this bitopological Gχ is a concrete functorial isomorphism. Now clearly
by the direct product of us-quantales, 22 ∼= 4, so by 3.23 and 3.24, 2-BiTop
concretely, fully, strictly embeds into 4-Top. Hence via the composition

E× ◦ Gχ : BiTop →֒ 4-Top,

BiTop concretely, fully, strictly embeds into 4-Top. For the monoreflectivity
claim, see 3.26 below; and the claim regarding limits and colimits follows from
Subsubsections 3.4.2–3.4.3 below, the limit claim needing the observation that
4 is a u-quantale with ⊗ = ∧. �



Bitopology as topology 93

3.4.2. Behavior of E× : L-BiTop → L
2-Top w.r.t. colimits. Since for all

consistent L the full concrete embedding E× is not a functorial isomorphism,
but only a strict embedding, it is worthwhile to investigate its behavior w.r.t.
limits and colimits. This subsection shows for any us-quantale L that the
embedding E× has a right-adjoint—and hence preserves colimits. The next
subsection then shows step by step for L a u-quantale that the Special Adjoint
Functor Theorem constructs for E× a left-adjoint—and hence E× preserves
limits; and further the next subsection shows for any us-quantale L that the
embedding E× reflects and detects all limits and is transportable. Therefore,
this subsection—in concert with the preceding and subsequent subsections—
shows that E× is an extremely well-behaved embedding.

Theorem 3.26 (E× ⊣ Fπ). Let L be a us-quantale and put the “projection”
functor Fπ : L-BiTop ← L

2-Top as follows:

Fπ (X, τ ) = (X, Fπ (τ )) , Fπ (f ) = f,

where the fibre level of Fπ

Fπ (τ ) = (π1 ◦ τ ≡{π1 ◦ u : u ∈ τ} , π2 ◦ τ ≡{π2 ◦ u : u ∈ τ})

uses the projections π1, π2 : L × L → L for the us-quantalic (direct) product.
Then the following hold:

(1) Fπ is a concrete embedding which is not full and does not lift limits.
(2) E× ⊣ Fπ , so E× preserves all strong colimits and Fπ preserves all

strong limits.
(3) Fπ need not detect limits nor be transportable.
(4) Fπ ◦ E× = IdL-BiTop.
(5) L-BiTop is isomorphic (via E×) to a full monocoreflective subcategory

of L2-Top.
(6) L2-Top is isomorphic (via Fπ) to an isoreflective subcategory of

L-BiTop.

Proof. Ad(1). Since us-quantalic projections preserve arbitrary joins, the ten-
sor, and the unit, it follows that Fπ (τ ) ∈ L-BT (X); and hence (X, Fπ (τ )) ∈
|L-BiTop| and Fπ is well-defined at the object level. As for morphisms, let
f : (X, τ ) → (Y, σ) be L2-continuous in L2-Top. Then, given v ∈ σ, the
identities

f←L (π1 ◦ v) = π1 ◦ f
←
L (v) , f

←
L (π2 ◦ v) = π2 ◦ f

←
L (v)

are easily checked and immediately imply that f : (X, Fπ (τ )) → (Y, Fπ (σ)) is
L-bicontinuous in L-BiTop. Now by the concreteness of Fπ, it is immediately
a concrete and faithful functor. To show that Fπ is an embedding, it remains
to check that Fπ injects objects: but if u, v : X → L

2 are distinct, there exists
x ∈ X such that W.L.O.G.

π1 (u (x)) 6= π1 (v (x)) ;

which implies that if τ 6= σ as L2-topologies on X, then Fπ (τ ) 6= Fπ (σ) as
L-bitopologies on X, showing that Fπ injects objects.



94 S. E. Rodabaugh

To see that Fπ need not be full, let L = 2, write the Boolean algebra L
2 = 4

as {(⊥,⊥) , (⊥,⊤) , (⊤,⊥) , (⊤,⊤)}, let X = {x} , and choose

τ =
{

(⊥,⊥), (⊤,⊤)
}

, σ =
{

(⊥,⊥), (⊥,⊤), (⊤,⊥), (⊤,⊤)
}

.

Then it follows idX : (X, τ ) → (X, σ) is not L
2-continuous (since σ is not a

subset of τ ). Now

π1 ◦ (⊥,⊥) = π1 ◦ (⊥,⊤) = ⊥, π1 ◦ (⊤,⊥) = π1 ◦ (⊤,⊤) = ⊤,

π2 ◦ (⊥,⊥) = π2 ◦ (⊤,⊥) = ⊥, π2 ◦ (⊥,⊤) = π2 ◦ (⊤,⊤) = ⊤,

so that

Fπ (τ ) = (π1 ◦ τ, π2 ◦ τ )

= ({⊥,⊤} , {⊥,⊤})

= (π1 ◦ σ, π2 ◦ σ)

= Fπ (σ) ,

implying idX : (X, Fπ (τ )) → (X, Fπ (σ)) is L-bicontinuous. The concreteness
of Fπ implies there exists no g ∈ L-Top with Fπ (g) = idX , so Fπ is not full.

To see that Fπ need not lift limits, let the diagram in L
2-Top be the space

(X, σ) of the preceding paragraph. Then the image of this diagram is the space
(X, Fπ (σ)) in L-BiTop. Now the space (X, Fπ (τ )), together with the arrow
idX : (X, Fπ (τ )) → (X, Fπ (σ)), is a limit of the diagram (X, Fπ (σ)): any
L-bicontinuous f : (Z, υ1, υ2) → (X, Fπ (σ)) trivially factors uniquely through
idX . But as seen in the preceding paragraph, there is no g ∈ L-Top with
Fπ (g) = idX , which means there is no limiting cone of (X, σ) in L

2-Top which
Fπ carries over to the limit idX : (X, Fπ (τ )) → (X, Fπ (σ)) in L-BiTop. Hence
Fπ need not lift limits.

Ad(2). Let (X, τ1, τ2) ∈ |L-BiTop| be given. Then

FπE× (X, τ1, τ2) = Fπ (X, ϕ
→
X (τ1 × τ2)) ≡ (X, τ̂1, τ̂2) ,

where it follows that

τ̂1 = {π1 ◦ ϕX (u, v) : u ∈ τ1, v ∈ τ2} ,

τ̂2 = {π1 ◦ ϕX (u, v) : u ∈ τ1, v ∈ τ2} .

We choose the right unit η to be the identity mapping id : X → X. Then for
each x ∈ X,

(π1 ◦ ϕX (u, v)) (x) = π1 (u (x) , v (x)) = u (x) ,

(π2 ◦ ϕX (u, v)) (x) = π2 (u (x) , v (x)) = v (x) ,

which immediately gives the L-bicontinuity of η.
For universality of the lifting, let (X, τ ) ∈

∣

∣L2-Top
∣

∣ be given, along with

an L-bicontinuous map f : (X, τ1, τ2) → (X, Fπ (τ )). Choosing f̄ = f , we now
check f̄ : E× (X, τ1, τ2) → (X, τ ) is an L-continuous map from E× (X, τ1, τ2)
to (X, τ ) by letting v ∈ τ and x ∈ X. Then the L-bicontinuity of f implies

f←L (π1 ◦ u) ∈ τ1, f
←
L (π2 ◦ u) ∈ τ2,



Bitopology as topology 95

from which it follows

ϕX (f
←
L (π1 ◦ u) ∈ τ1, f

←
L (π2 ◦ u) ∈ τ2) ∈ ϕ

→
X (τ1 × τ2) .

Further, we note

f←L (u) (x) = u (f (x)) = (π1 (u (x)) , π2 (u (x)))

= (f←L (π1 ◦ u) (x) , f
←
L (π2 ◦ u) (x))

= ϕX (f
←
L (π1 ◦ u) ∈ τ1, f

←
L (π2 ◦ u) ∈ τ2) (x) .

Finally, it is immediate that f̄ is the unique L-continuous map from E× (X, τ1, τ2)
to (X, τ ) such that

f = f̄ ◦ η,

completing the universality of the lifting. The naturality diagram now follows
by concreteness.

Ad(3). This is an immediate consequence of (1), (2), and Proposition 13.34
[1].

Ad(4). Since τ̂1 = τ1, τ̂2 = τ2 in the proof of (2), it is immediate that
Fπ ◦ E× = IdL-BiTop.

Ad(5). Using Fπ ◦E× = IdL-BiTop, the components of the left unit (counit)
of E× ⊣ Fπ furnish the needed monocoreflection arrows to L

2-topological spaces
from the E× image of L-BiTop.

Ad(6). Using Fπ ◦ E× = IdL-BiTop, the components of the right unit of
E× ⊣ Fπ furnish the needed isocoreflection arrows to L-bitopological spaces
from the Fπ image of L

2-Top. �

Remark 3.27. We collect some facts concerning E×, Fπ , and their fibre-
dependent constructions, where L is a us-quantale:

(1) Fπ ⊣/ E× if L is consistent. This is a consequence of 3.24.
(2) E× ⊣ Fπ need not be a categorical equivalence. This follows from (1).
(3) For each (X, τ1, τ2) ∈ |L-BiTop| ,

FπE× (τ1, τ2) = Fπ (ϕ
→
X (τ1 × τ2))

= (π1 ◦ ϕ
→
X (τ1 × τ2) , π2 ◦ ϕ

→
X (τ1 × τ2))

= (τ1, τ2) .

(4) For each (X, τ ) ∈
∣

∣L2-Top
∣

∣ ,

“E× (Fπ (τ )) = ϕ
→
X ((π1 ◦ τ ) × (π2 ◦ τ )) ⊃ τ ”

always holds; but for L consistent,

“E× (Fπ (τ )) = ϕ
→
X ((π1 ◦ τ ) × (π2 ◦ τ )) ⊂ τ ”

need not hold. The latter statement is another version of (1).

Theorem 3.28 (characterization of
∣

∣E→× (L-BiTop)
∣

∣). Let L be a us-quantale

and (X, τ ) ∈
∣

∣L2-Top
∣

∣. Then (X, τ ) ∈
∣

∣E→× (L-BiTop)
∣

∣ if and only if
E× (Fπ (τ )) = τ , i.e., both inequalities of 3.27(4) hold.



96 S. E. Rodabaugh

3.4.3. Behavior of E× : L-BiTop → L
2-Top w.r.t. limits. The question of a

left-adjoint for E× is open for general us-quantales L; and it is our conjecture is
that for general us-quantales L, E× would not preserve products or intersections
and hence would not have a left-adjoint. But on the other hand, this section
shows E× has a left-adjoint (and therefore preserves all limits) for L any u-
quantale. We point out that our proof of this left-adjoint is existential (via the
Special Adjoint Functor Theorem) and not constructive; and it is an additional
open question whether there is a direct construction of this left adjoint not
essentially factoring through our proof. It is further proved that E× reflects
and detects limits and is transportable.

Lemma 3.29 (preservation of products). For each u-quantale L, E× : L-
BiTop → L2-Top preserves arbitrary (small) products.

Sublemma 3.30. Let L be a u-quantale and suppose X is a set and τ1, τ2 are
L-topologies on X with respective subbases σ1, σ2, namely

τ1 = 〈〈σ1〉〉 , τ2 = 〈〈σ2〉〉 ,

such that {⊥, e}⊂ σ1 ∩ σ2. Then

(*) ϕ→X (τ1 × τ2) = 〈〈ϕ
→
X (σ1 × σ2)〉〉 .

Proof. To see that “⊃” holds in (*), note that

σ1 × σ2 ⊂ τ1 × τ2,

ϕ→X (σ1 × σ2) ⊂ ϕ
→
X (τ1 × τ2) ,

〈〈ϕ→X (σ1 × σ2)〉〉⊂ ϕ
→
X (τ1 × τ2) .

For “⊂” in (*), we first invoke the associativity of ⊗ and its infinite distribu-
tivity over

∨

to write members of τ1, τ2 as joins of tensor products of members
of σ1, σ2, respectively. More precisely, consider these typical members

∨

α∈A1





⊗

β ∈B1

uαβ



 ,
∨

α∈A2





⊗

β ∈B2

vαβ





of τ1, τ2, respectively, where A1, A2 are arbitrary indexing sets, B1, B2 are ar-
bitrary finite indexing sets, each uαβ ∈ σ1, each vαβ ∈ σ2, and where W.L.O.G.
we assume

A1 ∩ A2 = ∅ = B1 ∩ B2.

Next, we augment the uαβ ’s and vαβ ’s as follows, using the assumption that
{⊥, e}⊂ σ1 ∩ σ2:

α ∈ A1, β ∈ B2, uαβ ≡ e,

α ∈ A2, β ∈ B1 ∪ B2, uαβ ≡⊥,

α ∈ A2, β ∈ B1, vαβ ≡ e,

α ∈ A1, β ∈ B1 ∪ B2, vαβ ≡⊥.



Bitopology as topology 97

It follows that as maps from X to L that

∨

α∈A1 ∪A2





⊗

β ∈B1 ∪B2

uαβ



 =
∨

α∈A1





⊗

β ∈B1

uαβ



 ,

∨

α∈A1 ∪A2





⊗

β ∈B1 ∪B2

vαβ



 =
∨

α∈A2





⊗

β ∈B2

vαβ



 .

We thus have that a typical member




∨

α∈A1





⊗

β ∈B1

uαβ



 ,
∨

α∈A2





⊗

β ∈B2

vαβ









of τ1 × τ2 may be rewritten as




∨

α∈A1 ∪A2





⊗

β ∈B1 ∪B2

uαβ



 ,
∨

α∈A1 ∪A2





⊗

β ∈B1 ∪B2

vαβ









=
∨

α∈A1 ∪A2





⊗

β ∈B1 ∪B2

(uαβ , vαβ )



 ,

the latter being the form of a typical member of 〈〈σ1 × σ2〉〉 . To complete the
proof of “⊂”, we invoke the fact that ϕ→X is an order-isomorphism preserving
all tensor products (3.18(4)) to conclude that

ϕ→X (τ1 × τ2) ⊂ ϕ
→
X 〈〈σ1 × σ2〉〉 = 〈〈ϕ

→
X (σ1 × σ2)〉〉 .

�

Proof of 3.29. Recall the categorical products in L-BiTop use the categorical
product of L-Top in each slot as well as the usual projections for the morphisms
of the product (Subsection 1.5), and let {(Xγ , (τ

γ
1 , τ

γ
2 ))}γ ∈Γ ⊂ |L-BiTop|.

Because of the concreteness of E×, the validity of

E×





∏

γ ∈Γ

(Xγ , (τ
γ
1 , τ

γ
2 )) ,{πγ}γ ∈Γ



 =





∏

γ ∈Γ

E× (Xγ , (τ
γ
1 , τ

γ
2 )) ,{πγ}γ ∈Γ





holds if and only if we have the equality of topologies

(**) ϕ→×γ ∈ ΓXγ (Πγ ∈Γ τ
γ
1 × Πγ ∈Γ τ

γ
2 ) = Πγ ∈Γ ϕ

→
Xγ

(τ
γ
1 × τ

γ
2 ) ,

where “×” denotes as usual the direct product of us-quantales. For convenience,
“LHS” and “RHS” respectively denote the left-hand side and right-hand side
of (**). Let a subbasic open subset W be given from RHS. Then W may be
written as follows:

W = (πβ )
←
L

(

ϕXβ

(

t
β
1 , t

β
2

))

,



98 S. E. Rodabaugh

where
(

t
β
1 , t

β
2

)

∈ τ
β
1 × τ

β
2 for a fixed index β ∈ Γ. Given {xγ}γ ∈Γ ∈×γ ∈ΓXγ ,

then

W
(

{xγ}γ ∈Γ

)

= ϕXβ

(

t
β
1 , t

β
2

)(

πβ

(

{xγ}γ ∈Γ

))

= ϕXβ

(

t
β
1 , t

β
2

)

(xβ )

=
(

t
β
1 (xβ ) , t

β
2 (xβ )

)

=
([

(πβ )
←
L

(

t
β
1

)](

{xγ}γ ∈Γ

)

,
[

(πβ )
←
L

(

t
β
2

)](

{xγ}γ ∈Γ

))

.

This shows W is in LHS, LHS contains a subbasis of RHS, and so LHS contains
RHS.

For the reverse direction, let Z be in LHS. Then ∃ (u1, u2) ∈ Πγ ∈Γ τ
γ
1 ×

Πγ ∈Γ τ
γ
2 with Z = ϕ×γ ∈ ΓXγ (u1, u2) . Since the τ

γ
1 ’s and τ

γ
2 ’s contain {⊥, e}

and since these L-subsets are preserved by the Zadeh preimage operators of
all the projection maps, the usual subbasis for each of Πγ ∈Γ τ

γ
1 and Πγ ∈Γ τ

γ
2

contains {⊥, e}. Thus 3.30 applies to say it suffices to let u1, u2 be subbasic in
their respective L-product topologies

∏

γ ∈Γ τ
γ
1 ,
∏

γ ∈Γ τ
γ
2 ; so we may write

u1 = (πα)
←
L

(tα1 ) , u2 = (πβ )
←
L

(

t
β
2

)

,

where tα1 ∈ τ
α
1 , t

β
2 ∈ τ

β
2 for fixed indices α, β ∈ Γ. Let {xγ}γ ∈Γ ∈ ×γ ∈ΓXγ .

Then recalling that L has a unit e for ⊗ and that e is the corresponding unit
for ⊗ lifted to LX , we have

Z
(

{xγ}γ ∈Γ

)

= ϕ×γ ∈ ΓXγ (u1, u2)
(

{xγ}γ ∈Γ

)

=
(

u1

(

{xγ}γ ∈Γ

)

, u2

(

{xγ}γ ∈Γ

))

=
(

(πα)
←
L

(tα1 )
(

{xγ}γ ∈Γ

)

, (πβ )
←
L

(

t
β
2

)(

{xγ}γ ∈Γ

))

=
(

tα1

(

πα

(

{xγ}γ ∈Γ

))

, t
β
2

(

πβ

(

{xγ}γ ∈Γ

)))

=
(

tα1 (xα) , t
β
2 (xβ )

)

=
(

tα1 (xα) ⊗ e, e ⊗ t
β
2 (xβ )

)

= (tα1 (xα) , e (xα)) ⊗
(

e (xβ ) , t
β
2 (xβ )

)

=

(

[(πα)
←
L

(ϕXα (t
α
1 , e))]⊗

[

(πβ )
←
L

(

ϕXβ

(

e, t
β
2

))]

)

(

{xγ}γ ∈Γ

)

,

the last line being the evaluation at {xγ}γ ∈Γ by a tensor of open subsets of RHS

and hence of an open subset of RHS. Thus Z is in RHS, so LHS is contained
in RHS, completing the proof of the theorem. 2

Lemma 3.31. For each us-quantale L, E× : L-BiTop → L
2-Top preserves

equalizers.



Bitopology as topology 99

Sublemma 3.32. Let L be a us-quantale, (X, τ, σ) ∈ |L-BiTop| , Z ⊂ X, and
τ (Z) , σ (Z) , E× (τ, σ) (Z)be the L-subspace topologies on Z given by

τ (Z) =
{

u |Z : u ∈ τ
}

,

σ (Z) =
{

v |Z : v ∈ σ
}

,

E× (τ, σ) (Z) = ϕ
→
X (τ × σ) (Z)

(cf. [41]). Then

E× (τ, σ) (Z) = E× (τ (Z) , σ (Z)) .

Restated, E× respects subspace topologies.

Proof. Let u ∈ τ, v ∈ σ, z ∈ Z. Then

ϕX (u, v) |Z (z) = (u (z) , v (z)) =
(

u |Z (z) , v |Z (z)
)

= ϕX
(

u |Z , v |Z
)

(z) .

This implies

E× (τ, σ) (Z) = ϕ
→
X (τ × σ) (Z) = ϕ

→
X (τ (Z) × σ (Z)) = E× (τ (Z) , σ (Z)) .

�

Proof of 3.31. A categorical proof based upon the concreteness of E×, F and
FπE× = IdL-BiTop (3.26) does not work since it would generally require that
E×Fπ (τ ) ⊂ τ , which need not be true by (3.27(4)). It is necessary to look at
the actual construction of equalizers in each of L-BiTop and L2-Top and show
that E× carries the former into the latter. It can be checked that the equalizer of
f, g : (X, τ1, τ2) ⇉ (Y, σ1, σ2) in L-BiTop is given by ((Z, τ1 (Z) , τ2 (Z)) , →֒) ,
where

Z = {x ∈ X : f (x) = g (x)} ,

and that the equalizer of f, g : E× (X, τ1, τ2) ⇉ E× (Y, σ1, σ2) in L
2-Top is

given by ((Z, E× (τ1, τ2) (Z)) , →֒) using the same Z. Because of the concrete-
ness of E×, the issue is whether E× (τ1, τ2) (Z) is the same as E× (τ1 (Z) , τ2 (Z)),
and this is settled in 3.32. 2

Corollary 3.33. For each u-quantale L, E× : L-BiTop → L
2-Top preserves

all small limits. In particular, for each frame L, E× preserves all small limits.

Proof. It is not difficult to show that L-BiTop is topological over Set w.r.t. the
usual forgetful functor; and since Set is complete, it follows that L-BiTop is
complete (Theorem 21.16 [1]). Conjoin 3.29 and 3.31 to get that E× preserves
equalizers and (all) products; and then apply Proposition 13.4 [1] to finish the
proof. �

Lemma 3.34. For each u-quantale L, E× : L-BiTop → L
2-Top preserves all

intersections.

Proof. As in the proof of 3.31, it is necessary to look at the actual construction
of intersections in each of L-BiTop and L2-Top and show that E× carries the
former into the latter. Since this is trivially the case if the indexing class of the
intersection is empty, we assume sequens that the indexing class is nonempty.



100 S. E. Rodabaugh

To describe intersections in L-BiTop, let {((Xγ , τ
γ
1 , τ

γ
2 ) , mγ )}γ ∈Γ be a class

of subobjects of (Y, σ1, σ2)—by the well-poweredness of L-BiTop (Subsection
1.5), this class is not proper, i.e., we may take Γ as a set; form the product

(

(×γ ∈ΓXγ , Πγ ∈Γ τ
γ
1 , Πγ ∈Γ τ

γ
2 ) ,{πγ}γ ∈Γ

)

of these subobjects in L-BiTop; let

X ≡
{

{xγ}γ ∈Γ : ∀β, δ ∈ Γ, mβ (xβ ) = mδ (xδ)
}

⊂×γ ∈ΓXγ ;

fix ζ ∈ Γ; and put

m ≡ mζ ◦ πζ ◦ →֒ : X → Y .

Then equipping X with the L-subspace topologies

[Πγ ∈Γ τ
γ
1 ] (X) , [Πγ ∈Γ τ

γ
2 ] (X) ,

respectively, it can be shown that

((X, [Πγ ∈Γ τ
γ
1 ] (X) , [Πγ ∈Γ τ

γ
2 ] (X)) , m)

is the required intersection in L-BiTop.
We now consider in L2-Top the image

{(E× (Xγ , τ
γ
1 , τ

γ
2 ) , mγ )}γ ∈Γ = {((Xγ , E× (τ

γ
1 , τ

γ
2 )) , mγ )}γ ∈Γ ,

under E× of the family {((Xγ , τ
γ
1 , τ

γ
2 ) , mγ )}γ ∈Γ , which image by the functo-

riality of E× is a sink of subobjects for E× (Y, σ1, σ2). Using the X and m of
the preceding paragraph, it can be shown that

((X, [Πγ ∈Γ E× (τ
γ
1 , τ

γ
2 )] (X)) , m)

is the required intersection in L2-Top.
To show that E× takes the L-BiTop intersection to the L

2-Top intersection,
we note

E× ([Πγ ∈Γ τ
γ
1 ] (X) , [Πγ ∈Γ τ

γ
2 ] (X))

= E× (Πγ ∈Γ τ
γ
1 , Πγ ∈Γ τ

γ
2 ) (X) (by 3.32)

= [Πγ ∈Γ E× (τ
γ
1 , τ

γ
2 )] (X) (by proof of 3.29 (**)),

which shows

E× (X, [Πγ ∈Γ τ
γ
1 ] (X) , [Πγ ∈Γ τ

γ
2 ] (X)) = (X, [Πγ ∈Γ E× (τ

γ
1 , τ

γ
2 )] (X)) .

�

Theorem 3.35. For each u-quantale L, E× : L-BiTop → L
2-Top preserves

all strong limits.

Proof. This follows from 3.33, 3.34, and Definition 13.1(3) [1]. �

Theorem 3.36. For each u-quantale L, E× : L-BiTop → L
2-Top has a left

adjoint.



Bitopology as topology 101

Proof. First, L-BiTop has small fibres and is a topological construct (proof
of 3.33); hence, L-BiTop is complete and well-powered with coseparators by
Corollary 21.17 [1]. Second, Proposition 12.5 [1] now gives L-BiTop is strongly
complete. Third, since E× preserves all strong limits (3.35), the Special Adjoint
Functor Theorem 18.17 [1] now implies E× is a right-adjoint. Finally, apply
Proposition 18.9 [1]. �

Proposition 3.37. For each us-quantale L, E× : L-BiTop → L
2-Top reflects

and detects all limits and hence lifts all limits and is transportable.

Proof. The details are straightfoward using the preservation of limits by Fπ,
Fπ ◦ E× = IdL-BiTop, 3.36, and Proposition 13.34 [1]. �

3.4.4. Behavior of E× : L-BiTop → L
2-Top w.r.t. stratification issues. This

subsubsection shows E× is essentially neutral w.r.t. stratification issues.

Lemma 3.38. Let L be a us-quantale, (X, τ, σ) ∈ |L-BiTop|, and (γ, δ) ∈ L2.
Then (γ, δ) ∈ E× (τ, σ) if and only if γ ∈ τ and δ ∈ σ.

Proof. It is straightforward to check that

(γ, δ) ∈ ϕ→X (τ × σ) ⇔ ∃u ∈ τ, ∃v ∈ σ, ∀x ∈ X, (u (x) , v (x)) = (γ, δ)

⇔ γ ∈ τ, δ ∈ σ.

�

Theorem 3.39. Let L be a us-quantale,(X, T) ∈ |Top| , (X, τ ) ∈ |L-Top| ,
and (X, τ, σ) ∈ |L-BiTop|. The following hold:

(1) E× (X, τ, σ) always has (⊥,⊥), (⊥, e), (e,⊥), (e, e) as open subsets.

(2) E× (X, τ, σ) is anti-stratified if and only if L is inconsistent.
(3) For L = 2, E× (X, τ, σ)is weakly stratified.
(4) E×Gχ (X, T) is weakly stratified for |L| = 2 and non-stratified for
|L| > 2.

(5) E× (X, τ, σ) is weakly stratified if and only if (X, τ, σ) is weakly strati-
fied.

(6) E× (X, τ, σ) is non-stratified if and only if (X, τ, σ) is non-stratified.
(7) Statements (1–3, 5–6) with E× (X, τ, σ) replaced with E×Fd (X, τ ) and

(X, τ, σ) replaced with (X, τ ).

Proof. (1) follows from 3.38 given that {⊥, e}⊂ τ ∩ σ; (2, 3, 4) follow from (1)
and the fact that 4 may be taken as precisely {(⊥,⊥) , (⊥, e) , (e,⊥) , (e, e)};
(5) follows from 3.38; (6) contraposes (5); and (7) is immediate from the other
statements. �

4. Summary

This paper surveys the relationship between (lattice-valued) bitopology and
(lattice-valued) topology by examing a variety of functorial relationships
—Ed, Fl, Fr, F∧, F∨, FΠ, E×, Fπ —when L is a us-quantale. From this overview



102 S. E. Rodabaugh

of these functors and their properties, the following metamathematical conclu-
sions emerge:

(1) If it were assumed that the underlying lattice L of membership values
is not allowed to change, then this survey would support the following
viewpoint:
(a) (lattice-valued) bitopology is strictly more general than (lattice-

valued) topology in an extremely well-behaved way—justified by
Ed; and

(b) (lattice-valued) topology is not more general than (lattice-valued)
bitopology—justified by Fl, Fr, F∧, F∨, FΠ in comparison with Ed,
though the variety of ways in which bitopological spaces may be
interpreted as topological spaces is rather striking.

(2) If it were assumed that the underlying lattice L of membership values
is allowed to change (e.g., to the direct s-quantalic product L2), then
this survey would support the following viewpoint:
(a) (lattice-valued) topology is strictly more general than (lattice-

valued) bitopology in an extremely well-behaved way—justified
by E×; and

(b) (lattice-valued) bitopology is not more general than (lattice-valued)
topology—justified by Fπ in comparison with E×, though Fπ is
a rather interesting interpretation of topological spaces as bitopo-
logical spaces.

(3) This paper supports viewpoint (2) against viewpoint (1) for the follow-
ing reasons:
(a) We are in fact allowed to choose whatever underlying lattice of

membership values we wish, so in fact the underlying assumption
of (1) is false and the underlying assumption of (2) is true. The
class of embeddings E× stands and must be reckoned with.

(b) Topology (lattice-valued) is fundamentally simpler than bitopol-
ogy (lattice-valued):

(i) An L-bitopological space (X, τ, σ) adds to the ground object
X three parameters—L, τ, σ; while an M -topological space
(X, τ ) adds to the ground object X two parameters—M, τ .

(ii) When passing (via E×) from the L-bitopological space
(X, τ, σ) to the L2-topological space, the complexity of two
topologies is isolated in the underlying lattice of membership
values, leaving behind one topology.

(c) Topology (lattice-valued) is strictly more general than bitopology
(lattice-valued) in each of two ways:

(i) For each L ∈ |USQuant| , the direct product L2 ∈ |USQuant|
and L-BiTop embeds as a strict subcategory of L2-Top (via
E×), which is extremely well-behaved if L ∈ |UQuant| .

(ii) The class

{

L2-Top : L ∈ |USQuant|
}



Bitopology as topology 103

representing the field of fixed-basis bitopology using us-quantales
is a strictly proper subclass of the class

{L-Top : L ∈ |USQuant|}

representing the field of fixed-basis topology using us-quantales
(and not every us-quantale is a direct square of another us-
quantale), and this strictness holds if the class is indexed by
|UQuant| .

(iii) Thus when one proves a theorem in fixed-basis topology, it
is strictly more general w.r.t. coverage of categories and
coverage of objects in each category in which bitopological
spaces are embedded.

(d) The upshot of (a, b, c) is that (lattice-valued) bitopology is cate-
gorically redundant, particularly for underlying unital quantales:
(lattice-valued) topology is fundamentally simpler and strictly more
general. Fixed-basis bitopology is a complicated version of re-
stricted subcategories of categories from a restricted class of cate-
gories of fixed-basis topological spaces. For lattice-theoretic bases
larger than 2, workers in lattice-valued bitopology should now be
working in lattice-valued topology.

(4) The above arguments apply to traditional bitopology in a more sub-
tle way. On the one hand, traditional bitopology is isomorphic—in
an extremely well-behaved way—to a strictly proper, extremely well-
behaved subcategory of the much simpler 4-topology (BiTop embeds
into 4-Top: 3.25 above); restated, traditional bitopology is a restricted
subcase of a particular kind of fuzzy topology (namely 4-topology)
and therefore traditional bitopology is categorically redundant vis-a-vis
fixed-basis lattice-valued topology. On the other hand, the crisp lattice
2 underlying BiTop is so extremely simple that it is really a question
of two topologies in BiTop vis-a-vis the lattice 4 and one topology in
4-Top; restated, moving from (X, T, S) to (X, E× (T, S)) means mov-
ing from the parameters (2, T, S) to the parameters (4, E× (T, S)) ,
with the increased complexity in going from 2 to 4 offset by going
from the two topologies T, S to the one 4-topology E× (T, S) , noting
that each of T, S is more complex than 4. At the very least, workers
in traditional bitopology should consider working in 4-topology.

(5) The above arguments for redundancy in some sense are even stronger
than those used in [16] to show that various versions of “intuitionistic”
topologies or topologies comprising double subsets are redundant and a
categorically special case of fixed-basis topology since the E×’s of this
paper are strict embeddings and not functorial isomorphisms (when L
is consistent) as in [16].

(6) The rich history and literature of traditional bitopology, including in-
teresting separation and compactness axioms which “mix” together the
two topologies, are now immediately part of the literature of 4-Top



104 S. E. Rodabaugh

since the functorial embedding E×◦Gχ is an embedding at the power-
set and fibre levels in which these axioms are formulated. The precise
shape of these axioms as packaged by E×◦Gχ in 4-Top is, however, an
open question. Answering this question may teach us how to use suc-
cessful axioms of traditional bitopology to formulate successful axioms
for fixed-basis topology.

We illustrate (6) by showing that from traditional bicompactness E× in-
duces the compactness of [5] for lattice-valued topology and by discussing the
relationship between the respective Tihonov Theorems for the two categories
BiTop and 4-Top. As repeatedly shown in [36, 37, 38, 42, 34], Chang’s original
axiom of compactness [5] for lattice-valued topology, dubbed localic compact-
ness in [38] and simply compactness in [19, 42], has been extraordinarily
successful and justified with regard to classes of representations of L-spatial
locales, L-coherent locales, distributive lattices, Boolean algebras, traditional
compact Hausdorff spaces, classes of Stone-Čech compactifications, classes of
Stone-Weierstraß theorems [42], etc; indeed, for L a frame, only this compact-
ness axiom (and the very closely related axiom of [20]) has an unrestricted
compactification reflector for all of L-Top. Further, its Tihonov Theorem,
namely the Goguen-Tihonov Theorem [12], is one of the few Tihonov Theo-
rems in the fuzzy literature which does not need the classical theorem in its
proof; and hence it generalizes and explains both the statement and the proof
of the classical theorem. We need the statement of this theorem.

Let L be any complete lattice and let κ be a cardinal. We say ⊤ is κ-isolated
[12] in L if for each A ⊂ L −{⊤} with |A| ≤ κ,

∨

A < ⊤.

Theorem 4.1 (Goguen-Tihonov [12]). Let L be a complete lattice and Γ be
an indexing set. Then ⊤ is |Γ|-isolated in L if and only if each collection
{(Xγ , τγ ) : γ ∈ Γ} ⊂ L-Top of compact spaces (in the sense of [5]) yields a
compact product

∏

γ ∈Γ (Xγ , τγ ).

Corollary 4.2. The traditional Tihonov Theorem holds: for any indexing set
Γ,
∏

γ ∈Γ (Xγ ,Tγ ) is compact if and only if each (Xγ ,Tγ ) is compact.

Proof. The forward direction—the easier direction—can be given the usual
proof. As for the backward direction—the harder direction, we proceed as fol-
lows. First, the backward direction transfers directly, via the functorial isomor-
phism Gχ : Top → 2-Top, to the claim that each collection {(Xγ , τγ ) : γ ∈ Γ}⊂
2-Top of compact spaces (in the sense of [5]) yields a compact product
∏

γ ∈Γ (Xγ , τγ ); and this claim holds immediately from 4.1 since in the lat-
tice 2, ⊤ is κ-isolated in 2 for each cardinal κ, and so the claim holds for each
indexing set Γ. �

A traditional bicompact bitopological space (X, T, S) is defined by saying
that X is compact w.r.t. each of the topologies T, S. Given the construction of
products in BiTop (Subsection 1.5), we immediately have the usual Tihonov
Theorem for traditional bitopology.



Bitopology as topology 105

Corollary 4.3. For any indexing set Γ,
∏

γ ∈Γ (Xγ , Tγ , Sγ ) is bicompact if

and only if each (Xγ , Tγ , Sγ ) is bicompact.

Corollary 4.4. Let Γ be an indexing set. Then each collection {(Xγ , τγ ) : γ ∈ Γ}
⊂ 4-Top of compact spaces (in the sense of [5]) yields a compact product
∏

γ ∈Γ (Xγ , τγ ) if and only if |Γ| = 0 or 1.

Proof. Letting 4 be written as {⊥, a, b,⊤} with a, b unrelated, this is immediate
from 4.1 since ⊤ is κ-isolated in 4 if and only if κ ≤ 1. �

The plot thickens with the next definition, theorem, and corollary.

Definition 4.5. Let L ∈ |USQuant| . An L-bitopological space (X, τ1, τ2) is
(L-)bicompact if X is compact (in the sense of [5]) w.r.t. each of τ1 and τ2.

Theorem 4.6. For each L ∈ |USQuant|, E× : L-BiTop → L
2-Top preserves

bicompactness to compactness in the sense of [5].

Proof. Let a bicompact L-topological space (X, τ1, τ2) be given and let

{uγ × vγ : γ ∈ Γ}

be a cover of X from the L2-topology E× (τ1, τ2). If Γ is finite, then this cover
is its own finite subcover; so we assume Γ is not finite. Now

(⊤,⊤) = (⊤,⊤) =
∨

γ∈Γ

(uγ × vγ ) =
∨

γ∈Γ

uγ ×
∨

γ∈Γ

vγ ,

forcing each of {uγ : γ ∈ Γ} and {vγ : γ ∈ Γ} to be covers of X from τ1 and
τ2, respectively. The bicompactness yields two finite subcovers which we may
respectively write as follows:

{ui : i = 1, ..., m} , {vi : i = m + 1, ..., m + n} .

Then |Γ| ≥ m + n and

m+n
∨

i=1

(ui × vi) =

m+n
∨

i=1

ui ×

m+n
∨

i=1

vi ≥

m
∨

i=1

ui ×

m+n
∨

i=m+1

vi = ⊤×⊤ = (⊤,⊤),

showing that {ui × vi : i = 1, ..., m + n} is the needed subcover of X. �

Corollary 4.7. The functorial embedding E×◦Gχ : BiTop → 4-Top preserves
bicompactness to compactness in the sense of [5].

Proof. Since Gχ : BiTop → 4-BiTop preserves traditional bicompactness to
the bicompactness of 4.5, the corollary follows from 4.6. �

We close this discussion of (6) above with a few comments. First, tradi-
tional bicompactness mandates the compactness of [5] for lattice-valued topol-
ogy (4.7). Second, we note (E×Gχ)

→
(BiTop) is isomorphic to BiTop and

closed under all products (in 4-Top) (3.23, 3.29): this means that the cardi-
nality unrestricted Tihonov Theorem for BiTop (4.3) transfers to a cardinal-
ity unrestricted Tihonov Theorem for the subcategory (E×Gχ)

→
(BiTop) of

4-Top w.r.t. the compactness of [5]. Third, it now follows (4.4, 4.7) that E×



106 S. E. Rodabaugh

is not object-onto (already known) and that the special cardinality restriction
of the Goguen-Tihonov Theorem for 4-Top resides outside the subcategory
(E×Gχ)

→
(BiTop).

Acknowledgements. The referees are thanked for their thorough reading
and helpful criticisms which significantly improved this paper. The author is
especially grateful to the referee who found an important error in the original
submission following Corollary 4.4, the correction of which sheds light on the
topological properties of the E× functor.

References

[1] J. Adámek, H. Herrlich, G. E. Strecker, Abstract And Concrete Categories: The Joy
Of Cats, Wiley Interscience Pure and Applied Mathematics (John Wiley & Sons, Bris-
bane/Chicester/New York/Singapore/Toronto, 1990).

[2] B. Banaschewski, G. C. L. Brümmer, K. A. Hardie, Biframes and bispaces, Proc. Symp.
Categorical Algebra and Topology (Cape Town, 1981), Quaestiones Mathematicae 6
(1983), 13–25.

[3] J. Cerf, Groupes d’homotopie locaux et groupes d’homotopie mixtes des espaces
bitopologiques: Définitions et propriétés, C. R. Acad. Sci. Paris 252 (1961) 4093–4095.

[4] J. Cerf, Groupes d’homotopie locaux et groupes d’homotopie mixtes des espaces
bitopologiques: Presque n-locale connexion: Applications, C. R. Acad. Sci. Paris 253
(1961) 363–365.

[5] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182–190.
[6] C. De Mitri, C. Guido, G-fuzzy topological spaces and subspaces, Suppl. Rend. Circolo

Matem. Palermo 29 (1992), 363–383.
[7] C. De Mitri, C. Guido, Some remarks on fuzzy powerset operators, Fuzzy Sets and

Systems 126 (2002), 241–251.
[8] C. De Mitri, C. Guido, R. E. Toma, Fuzzy topological properties and hereditariness,

Fuzzy Sets and Systems 138 (2003), 127–147.
[9] J. T. Denniston, S. E. Rodabaugh, Functorial relationships between lattice-valued topol-

ogy and topological systems, in submission.
[10] A. Frascella, C. Guido, Structural lattices and ground categories of L-sets, in [24], 53–54.
[11] A. Frascella, C. Guido, Topological categories of L-sets and (L, M)-topological spaces

on structured lattices; in [13], 50.
[12] J. A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973), 734–742.
[13] S. Gottwald, P. Hájek, U. Höhle, E. P. Klement, eds, Fuzzy Logics And Related Struc-

tures: Abstracts, 26th Linz Seminar On Fuzzy Set Theory, February 2005, Bildungszen-
trum St. Magdalena, Linz, Austria, (Universitätsdirektion Johannes Kepler Universtät,
A-4040, Linz, 2005).

[14] C. Guido, The subspace problem in the traditional point set context of fuzzy topology,
in [26], 351–372.

[15] C. Guido, Powerset operators based approach to fuzzy topologies on fuzzy sets, Chapter
15 in [47], 401–413.

[16] J. Gutiérrez Garćıa, S. E. Rodabaugh, Order-theoretic, topological, categorical redun-
dancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic”
sets, “intuitionistic” fuzzy sets and topologies, Fuzzy Sets and Systems 156 (2005),
445–484.



Bitopology as topology 107

[17] U. Höhle, Many Valued Topology And Its Applications to Fuzzy Subsets, (Kluwer Aca-
demic Publishers, Boston/Dordrecht/London, 2001).

[18] U. Höhle, E. P. Klement, eds, Nonclassical Logics And Their Applications, Theory and
Decision Library—Series B: Mathematical and Statistical Methods, Volume 32 (Kluwer
Academic Publishers, Boston/Dordrecht/London, 1995).

[19] U. Höhle, S. E. Rodabaugh, eds, Mathematics Of Fuzzy Sets: Logic, Topology, And
Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3 (Kluwer Academic
Publishers, Boston/Dordrecht/London, 1999).

[20] U. Höhle, A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, Chapter 3 in
[19], 123–272.

[21] P. T. Johnstone, Stone Spaces, (Cambridge University Press, Cambridge, 1982).
[22] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13:3 (1963), 71–89.
[23] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic, Studia Logica

Library, Volume 8 (Kluwer Academic Publishers, Dordrecht/Boston/London, 2000).
[24] E. P. Klement, E. Pap, eds, Mathematics of Fuzzy Systems: Abstracts, 25th Linz Seminar

On Fuzzy Set Theory, February 2004, Bildungszentrum St. Magdalena, Linz, Austria
(Universitätsdirektion of Johannes Kepler Universität, A-4040, Linz, 2004).

[25] R. Kopperman, J. D. Lawson, Bitopological and topological ordered k-spaces, Topology
and its Applications 146–147 (2005), 385–396.

[26] W. Kotzé, ed, Special Issue, Quaestiones Mathematicae 20(3) (1997).
[27] S. S. Kumar, Weakly continuous mappings in fuzzy bitopological spaces, Proc. First

Tamilnadu Science Congress Tiruchirappalli (1992), 1–7.
[28] S. Mac Lane, Categories for the Working Mathematician (Springer-Verlag, Berlin),

1971.
[29] MathSciNet: Mathematical Reviews On The Web, database under “Anywhere =

bitopolog***”, 2006.
[30] M. E. Abd El-Monsef, A. E. Ramadan, α-Compactness in bifuzzy topological spaces,

Fuzzy Sets and Systems 30 (1989), 165–173.
[31] L. Motchane, Sur la notion d’espace bitopologique et sur les espaces de Baire, C. R.

Acad. Sci. Paris 244(1957), 3121–3124.
[32] L. Motchane, Sur la caractérisation des espaces de Baire, C. R. Acad. Sci. Paris 246

(1958), 215–218.
[33] C. J. Mulvey, M. Nawaz, Quantales: quantal sets, Chapter VII in [18], 159–217.
[34] A. Pultr, S. E. Rodabaugh, Lattice-valued frames, functor categories, and classes of

sober spaces, Chapter 6 in [47], pp. 153–187.
[35] A. Pultr, S. E. Rodabaugh, Category theoretic properties of chain-valued frames: Part

I: categorical and presheaf theoretic foundations, Fuzzy Sets and Systems, to appear;
Category theoretic properties of chain-valued frames: Part II: applications to lattice-
valued topology, Fuzzy Sets and Systems, to appear.

[36] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991),
297–345.

[37] S. E. Rodabaugh, Categorical frameworks for Stone representation theorems, Chapter
7 in [48], 178-231.

[38] S. E. Rodabaugh, Applications of localic separation axioms, compactness axioms, rep-
resentations, and compactifications to poslat topological spaces, Fuzzy Sets and Systems

73 (1995), 55–87.
[39] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic

(poslat) fuzzy set theories and topologies, in [26], 463–530.
[40] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topolo-

gies, Chapter 2 in [19], 91–116.
[41] S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, Chapter 4 in

[19], 273–388.
[42] S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and com-

pactifications, Chapter 7 in [19], 481–552.



108 S. E. Rodabaugh

[43] S. E. Rodabaugh, Fuzzy real lines and dual real lines as poslat topological, uniform, and
metric ordered semirings with unity, Chapter 10 in [19], 607–631.

[44] S. E. Rodabaugh, Axiomatic foundations for uniform operator quasi-uniformities,
Chapter 7 in [47], 199–233.

[45] S. E. Rodabaugh, Relationship of algebraic theories to powersets over objects in Set
and Set × C, to appear.

[46] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy
topological theories for lattice-valued mathematics, International Journal of Mathe-
matics and the Mathematical Sciences, 2007 (3) (2007), Article ID 43645, 71 pages.
doi:10.1155/2007/43645.
http://www.hindawi.com/getarticle.aspx?doi=10.1155/2007/43645.

[47] S. E. Rodabaugh, E. P. Klement, eds, Topological And Algebraic Structures In Fuzzy
Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in
Logic 20 (Kluwer Academic Publishers, Boston/Dordrecht/London, 2003).

[48] S. E. Rodabaugh, E. P. Klement, U. Höhle, eds, Application Of Category Theory To
Fuzzy Sets, Theory and Decision Library—Series B: Mathematical and Statistical Meth-
ods 14 (Kluwer Academic Publishers (Boston/Dordrecht/London, 1992).

[49] K. I. Rosenthal, Quantales And Their Applications, Pitman Research Notes In Mathe-
matics 234 (Longman, Burnt Mill, Harlow, 1990).

[50] A. S. M. Abu Safiya, A. A. Fora, M. W. Warner, Compactness and weakly induced fuzzy
bitopological spaces, Fuzzy Sets and Systems 62( 1994), 89–96.

[51] R. Srivastava, M. Srivastava, On compactness in bifuzzy topological spaces, Fuzzy Sets
and Systems 121 (2001), 285–292.

[52] C. K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45
(1974), 512–521.

[53] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338–353.

Received October 2006

Accepted April 2007

S. E. Rodabaugh (rodabaug@math.ysu.edu)
Department of Mathematics and Statistics, Youngstown State University, Youngstown,
Ohio 44555-3609, USA