@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 223–241

Extensions of closure spaces

D. Deses
∗
, A. De Groot-Van der Voorde, E. Lowen-Colebunders

Dedicated to Professor S. Naimpally on the occasion of his 70th birthday.

Abstract. A closure space X is a set endowed with a closure
operator P(X) →P(X), satisfying the usual topological axioms, except
finite additivity. A T1 closure extension Y of a closure space X induces
a structure γ on X satisfying the smallness axioms introduced by H.
Herrlich [?], except the one on finite unions of collections. We’ll use the
word seminearness for a smallness structure of this type, i.e. satisfying
the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show
that every T1 seminearness structure γ on X can in fact be induced by
a T1 closure extension. This result is quite different from its topological
counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield
in [?]. Also in the topological setting the existence of (strict) extensions
satisfying higher separation conditions such as T2 and T3 has been
completely characterized by means of concreteness, separatedness and
regularity [?]. In the closure setting these conditions will appear to be
too weak to ensure the existence of suitable (strict) extensions. In this
paper we introduce stronger alternatives in order to present internal
characterizations of the existence of (strict) T2 or strict regular closure
extensions.

2000 AMS Classification: 54A05, 54D35, 54E15, 54E17, 54D10.
Keywords: closure space, seminearness, separation, regularity, (strict) exten-
sion, minimal small stack.

1. Introduction.

The structures we will be dealing with, can be defined in various equivalent
ways, from which we shall use frequently two particular descriptions, namely
by small collections and by uniform covers.

∗The first author is ‘aspirant’ of the F.W.O.-Vlaanderen.



224 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

1.1. Let X be a set. A nonempty collection A ⊂ P(X), not containing ∅
is said to be a stack if A ∈ A whenever there exists B ∈ A with B ⊂ A. If
A⊂P(X) is an arbitrary collection of nonempty subsets then we put

stack A = {A ⊂ X|∃B ∈A : B ⊂ A}

and
sec A = {A ⊂ X|∀B ∈A : A∩B 6= ∅}

We write ẋ for stack {{x}}. In [?] H. Herrlich considered the following smallness
axioms, which can be expressed in terms of stacks in the following way. For
γ ⊂P2(X), a collection of stacks, consider the conditions

(S1) If A⊂B and A∈ γ then B ∈ γ
(S2) ∀x ∈ X : ẋ ∈ γ
(S3) γ 6= P2(X)
(S4) If (A∪B) ∈ γ then A∈ γ or B ∈ γ
(S5) If sec{cl A|A ∈A}∈ γ then secA∈ γ where cl A = {x ∈ X|sec {A,{x}}∈ γ}
A structure γ satisfying (S1),(S2),(S3) is called a prenearness structure, if

(S4) is added γ is a merotopic structure and if γ satisfies all five of the conditions
then it is a nearness structure. We will not be dealing with axiom (S4) but
we will assume that γ satisfies (S1), (S2), (S3) and (S5). Then γ is called a
seminearness structure and (X,γ) is a seminearness space. As in [?] a function
f : (X,γ) → (X′,γ′) between seminearness spaces is said to be uniformly
continuous if it preserves smallness in the sense that

A∈ γ ⇒ stack {f(A)|A ∈A}∈ γ′

Seminearness spaces and uniformly continuous maps form a topological con-
struct in the sense of [?]. We refer to the original papers [?], [?] for a systematic
study of prenearness and nearness spaces. On the latter a selfcontained text-
book ”Uniforme Raüme” appeared [?]. Another textbook by G. Preuss [?] also
contains an introduction to nearness spaces and to some of the more general
structures such as prenearness and merotopic structures. In [?] however the
latter are called seminearness spaces, so that the terminology used in [?] differs
from the one we use here.

1.2. In [?] an equivalent way of describing the structure was presented in terms
of uniform covers.

If X is a set then the following conditions on µ ⊂P2(X) are considered
(U1) If U ≺ V and U ∈ µ then V ∈ µ (where ≺ denotes the classical refine-

ment relation)
(U2) If U ∈ µ then U is a cover of X
(U3) ∅ 6= µ 6= P2(X)
(U4) If U ∈ µ and V ∈ µ then {U ∩V |U ∈U,V ∈V}∈ µ
(U5) If U ∈ µ then {intµU|U ∈U}∈ µ where intµU = {x ∈ X|{U,X −{x}}∈ µ}
The covers in µ are called uniform covers. In our setting we will not be

dealing with (U4), so our covering structures µ satisfy (U1), (U2), (U3) and



Extensions of closure spaces 225

(U5). (X,µ) then forms an equivalent way for the description of a seminearness
space and the translation between (X,γ) and (X,µ) is as usual:

U ∈ µ ⇐⇒ ∀A∈ γ : U ∩A 6= ∅

A∈ γ ⇐⇒ ∀U ∈ µ : U ∩A 6= ∅

1.3. Some of the examples we will construct in the last section of the pa-
per, satisfy even stronger conditions than the seminearness axioms. A uniform
space, described in terms of covers satisfies (U1), (U2), (U3), (U4) and the con-
dition (U5’), saying that every uniform cover has a uniform star refinement,
which is in fact stronger than (U5). If we leave out (U4), as we did before, and
retain (U1), (U2), (U3) and (U5’) then we still have a (covering) seminearness
space. In this case we will say that the seminearness space is a uniform semin-
earness space . A special case of this situation is the following. Let X be a set
and let

{Ui|i ∈ I}
be any collection of partitions of X then

U ∈ µ ⇐⇒ ∃i ∈ I : Ui ≺U

defines a (covering) uniform seminearness structure on X. It is said to be zero
dimensional since it is generated by a collection of partitions. These structures
are investigated in more detail in [?].

1.4. A closure space (X,C) is a pair, where X is a set and C is a subset of
the power set P(X) satisfying the conditions that X belongs to C and that C
is closed for arbitrary unions. The sets in C are called open sets. A function
f : (X,C) → (Y,D) between closure spaces (X,C) and (Y,D) is said to be con-
tinuous if f−1(D) ∈C whenever D ∈D. Cl is the construct with closure spaces
as objects and continuous maps as morphisms. Some isomorphic descriptions
of Cl are often used f.i. by giving the collection of all closed sets (the so called
Moore family [?]) where, as usual, the closed sets are the complements of the
open ones and continuity is defined accordingly. Another isomorphic descrip-
tion is obtained by means of a closure operator [?]. The closure operation
cl : P(X) → P(X) associated with a closure space (X,C) is defined in the
usual way by x ∈ cl A ⇐⇒ (∀C ∈ C : x ∈ C ⇒ C ∩ A 6= ∅) where A ⊂ X
and x ∈ X. This closure need not be finitely additive, but it does satisfy the
conditions cl ∅ = ∅, (A ⊂ B ⇒ cl A ⊂ cl B),A ⊂ cl A and cl(cl A) = cl A
whenever A and B are subsets of X. Continuity is then characterized in the
usual way. Finally closure spaces can also be equivalently described by means of
neighborhood collections of the points. These neighborhood collections satisfy
the usual axioms, except for the fact that the collections need not to be filters.
So in a closure space the neighborhood collection V(x) of a point x is a stack,
where every V ∈V(x) contains x and V(x) satisfies the open kernel condition.
In the sequel we will just write X for a closure space and we’ll choose the most
convenient form for its explicit structure.



226 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

Motivations for considering closure spaces can be found in several applica-
tions. We refer to [?] and [?] for applications in geometry, to [?] for applications
in lattice theory, to [?], [?] and [?] for the use of closures in the development of
representations of physical systems, to [?] for the use in social sciences and to [?]
for applications in the context of knowledge representation. The introduction
of [?] contains some more details on motivation.

A closure space satisfies the R0 symmetry axiom if x ∈ cl {y} ⇐⇒ y ∈ cl {x},
∀x,y ∈ X and it satisfies T1 if {x} is closed for every x ∈ X. If (X,γ) is a
seminearness space then the closure defined in paragraph 1.1 by

cl A = {x ∈ X|sec {A,{x}}∈ γ}

is an R0 closure in our sense. This closure is the underlying closure of (X,γ)
and we also say that it is compatible with (X,γ). Whenever we consider neigh-
borhood collections Vγ(x), convergence or open sets for a seminearness space
(X,γ), we are in fact referring to the underlying closure. As for nearness spaces
we have in this more general context that the neighborhood collections Vγ(x)
are minimal small stacks (where minimality refers to the inclusion order). Next
we further illustrate the relation between R0 closure spaces and seminearness
spaces.

1.5. Let Y be an R0 closure space and define a seminearness structure by

A∈ γ ⇐⇒ ∃y ∈ Y : V(y) ⊂A

Remark that the underlying closure of the seminearness γ coincides with the
given closure on Y . The construct of R0 closure spaces is bicoreflectively em-
bedded in the construct of seminearness spaces, cfr. [?], [?].

1.6. Let Y be an R0 closure space and let X be a subset of Y . The closure
structure of Y induces a seminearness structure on X as follows

A∈ γ ⇐⇒ ∃y ∈ Y : V(y) ⊂ stackYA

The underlying closure of γ on X coincides with the closure structure induced
by Y on X.

If X is dense in Y (clY X = Y ) then Y is said to be a closure extension of
X and we say that (X,γ) is induced by the extension Y of X. X is said to be
strictly dense in Y if {clY B|B ⊂ X} is a base for the closed subsets of Y , in
the sense that every closed set of Y can be obtained by intersecting sets from
the base. In that case Y is said to be a strict extension of X and (X,γ) is said
to be induced by a strict extension.

The meaning of 1.6 is that, given an R0 closure extension Y of a closure
space X, a seminearness structure γ is induced on X which is compatible with
the given closure on X.

The first question we will be dealing with in this paper is, whether every
seminearness γ compatible with X as a closure space, can be induced by some
R0 closure extension Y of X.



Extensions of closure spaces 227

The parallel question in the setting of topological spaces is whether every
compatible nearness space can be induced by some R0 topological extension.
This question was answered negatively by S.A. Naimpally and J.H.M. Whitfield
in [?].

A thorough study on extensions of topological spaces was later carried out
by H.L. Bentley and H. Herrlich in [?], in particular giving internal character-
izations for nearness spaces to be induced by T1, T2 or T3 (strict) extensions.

In this paper we’ll deal with the closure counterparts of such questions.

2. Extensions.

In this section, starting from a seminearness space (X,γ) we construct two
types of enlargements, one type are the so called ”loose” enlargements and the
other type is a strict one.

2.1. Construction of a loose enlargement. Let (X,γ) be a seminearness
space and let {yA|A∈ α} be a collection of points, not belonging to X and in
one to one correspondence to a collection α of nonconvergent small stacks with
an open base. Let X′ = X ∪{yA|A∈ α}. On X′ we define a closure structure
cl′ by determining the neighborhood collections of the points as follows:

V′(x) = stackX′Vγ(x) for x ∈ X
V′(yA) = stackX′A∩ ẏA for A∈ α

Clearly (X′,cl′) is an R0 closure space of which X is a dense subset. Moreover
if D is a stack on X and stackX′D converges in X′, then D is small in (X,γ).
It is clear that in order to obtain an extension of (X,γ) in the sense of 1.6 the
condition D ∈ γ ⇒∃A∈ α : A⊂D has to be fulfilled.

The following proposition is relevant in this respect since it shows that in
fact openbased stacks determine the structure.

Proposition 2.1. If (X,γ) is a seminearness space then for every A∈ γ there
exists a B ∈ γ such that B has an open base and such that B ⊂A.

Proof. For A ∈ γ let B = stack {B ⊂ X|B open,B ∈ A}. If U ∈ µ then so is
{int U|U ∈U}. So finally B∩U 6= ∅. �

Proposition 2.2. Let (X,γ) be a seminearness space and let

α = {A∈P2(X)|A is a small openbased nonconvergent stack}
Then (X′,γ′) is an R0 closure extension of (X,γ).

We conclude from this fact that every seminearness space can be induced
by an R0 closure extension.

Remark that by exactly the same construction one has that every T1 semin-
earness space is induced by some T1 extension. Remark also that these results
deviate from their well known topological counterparts, cfr. [?], [?], [?].

In general, given a T1 space (X,γ) there can be many different T1 extensions
inducing γ. On the other hand, as we will see, strict T1 extensions need not
exist. However, if there exist T1 strict extensions, then they are essentially



228 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

unique. The reason for this is explained in the next results on minimal small
stacks, where again minimality refers to the inclusion order on stacks.

The following result is quite parallel to its topological counterpart developed
in [?].

Proposition 2.3.
(1) If (X,γ) is a seminearness space induced by an R0 extension Y of X,

then every minimal small stack is a trace V(y)|X for some y ∈ Y .
(2) If (X,γ) is a seminearness space induced by a strict R0 extension Y of

X then
{V(y)|X|y ∈ Y}

is the collection of all minimal small stacks.

Proof.
(1) If M is minimal small and stackYM converges to y ∈ Y then V(y)|X ⊂
M and hence V(y)|X = M.

(2) Let y ∈ Y be some point of a strict extension Y , and suppose that
A ⊂ V(y)|X and that A is small. Suppose that stackYA converges to
z, i.e. V(z)|X ⊂A. It follows that z ∈ clY{y}. Indeed otherwise there
would exist a subset B ⊂ X such that y ∈ clY B and z 6∈ clY B. And
this is impossible. Hence we can conclude that V(y) = V(z) and so
A = V(y)|X.

�

Corollary 2.4. If (X,γ) is a seminearness space and Y is a strict T1 extension
then the points of Y are in one to one correspondence to the minimal small
stacks in γ.

Also the following construction is quite similar to its topological counterpart
[?],[?].

2.2. Construction of a strict enlargement. Let (X,γ) be a seminearness
space and let

X̂ = X ∪{yM|M nonconvergent minimal small}

where again different points yM are chosen to be outside of X and in one to
one correspondence with the minimal small nonconvergent stacks.

For A ⊂ X put
O(A) = int A∪{yM|A ∈M}

and let ĉl on X̂ be the closure having as an open base {O(A)|A ⊂ X}.
Remark that {ĉl K|K ⊂ X} is a base for the closed sets where ĉl K =

cl K ∪{yM|K ∈ sec M}.
Clearly (X̂, ĉl) is an R0 closure space and it contains X as a strictly dense

subset. Moreover if D is a stack on X and stack
X̂
D converges in X̂ then:

Either V
X̂

(yM) ⊂ stack D for M minimal small and not convergent, then we



Extensions of closure spaces 229

have M ⊂ D. Or V
X̂

(x) ⊂ stack D and then VX(x) ⊂ D. So in any case
D ∈ γ.

In order to obtain an extension of (X,γ) we need to impose the following
condition (cfr. [?]).

Definition 2.5. A seminearness space is concrete if the minimal small stacks
determine the structure in the following sense

∀A∈ γ : ∃M minimal small M⊂A

Proposition 2.6. (X,γ) is induced by a strict R0 extension if and only if it
is a concrete seminearness space.

Proof. If (X,γ) is a concrete seminearness space we make the construction
developed in paragraph 2.2 and we prove that (X̂, ĉl) is an extension. So it
remains to show that if D is a small stack on X it converges in (X̂, ĉl). Choose
M⊂D minimal small.
Either M = V(x) for some x ∈ X and then for A ⊂ X with x ∈ int A we have
int A ∈D and hence O(A) ∈D. So we have V

X̂
(x) ⊂D.

Or M does not converge. Then we prove that

stack
X̂
D ⊃V

X̂
(yM)

Let A ⊂ X such that yM ∈ O(A), i.e. A ∈ M. In view of proposition 2.1 the
stack M has an open base. Then also int A ∈ M and finally int A ∈ D. So
again we can conclude that O(A) ∈ stack

X̂
D.

Conversely, suppose that (X,γ) has a strict R0 extension (Y,clY ). Let D
be small in (X,γ) then stackYD converges to some y ∈ Y . Then clearly
D ⊃VY (y)|X and in view of proposition 2.3 we have that VY (y)|X is a minimal
small stack. Hence (X,γ) is concrete. �

Remark that using exactly the same construction one has that (X,γ) is
induced by a strict T1 closure extension if and only if it is T1 and concrete.
Remark that if Y is a strict T1 extension of (X,γ) then Y is unique up to an
isomorphism leaving X pointwise fixed. It can easily be seen that the function
φ : Y → X̂ mapping y ∈ Y to yM with M = VY (y)|X if y 6∈ X and mapping
x ∈ X to x, is bijective and satisfies φ(clY B) = ĉl B, for every B ⊂ X.
Therefore we also have φ(clY Z) = ĉl(φ(Z)) for every Z ⊂ Y .

The previous results on R0 and T1 strict extensions are completely analogous
to their topological counterparts. In the next section, where higher separation is
considered, the parallelism with the topological situation does not go through.

3. Separation and extensions.

In this section we introduce higher separation conditions for seminearness
spaces. The notion ”separatedness” was introduced in [?] in the setting of
prenearness spaces and it proved to be very useful in the study of topological
extensions. However, in our setting, in order to produce Hausdorff closure



230 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

extensions, ”separatedness” will no longer be strong enough. We briefly recall
some definitions and results.

If (X,γ) is a seminearness space then a stack A is said to be near if sec A is
small. For instance, if

⋂
A∈AclγA 6= ∅ then A is near. A stack A is said to be

concentrated if it is small and near. For example, the neighborhood collections
in (X,γ) are concentrated. Small filters are also always concentrated.

Definition 3.1. [?] A seminearness space (X,γ) is separated if for every con-
centrated stack A also

B = {B ⊂ X|stackX{B}∪A near}
is near.

The proof of the following proposition is similar to the one of proposition
10.5 in [?] and can be found in [?].

Proposition 3.2. For a seminearness (X,γ) the following are equivalent
(1) (X,γ) is separated
(2) Every concentrated stack contains a unique minimal small stack

Proposition 3.3. If (X,γ) is separated, M is a minimal small concentrated
stack and M6= V(x) then

∃A ∈V(x) : ∃M ∈M : M ∩A = ∅

Proof. If on the other hand every A ∈ V(x) intersects every M ∈ M then
V(x) ∪M would be concentrated and then V(x) = M in view of the previous
proposition. �

Corollary 3.4. If (X,γ) is separated and T1 then in the underlying closure,
distinct points have disjoint neighborhoods. In particular a closure space (con-
sidered as a seminearness space) is separated and T1 if and only if distinct
points have disjoint neighborhoods. We use the label ”Hausdorff” or T2 for this
property.

The following conditions (i) and (ii) clearly are strengthening those formu-
lated in proposition 3.3.

Proposition 3.5. For a seminearness space (X,γ) the following are equivalent
(i) for M and N minimal small concentrated stacks, M6= N then

∃M ∈M : ∃N ∈N : N ∩M = ∅
(ii) for M and N minimal small stacks, M6= N then

∃M ∈M : ∃N ∈N : N ∩M = ∅

Proof. That (i) implies (ii) follows from the observation that when N is small
but not concentrated, then sec N is not small. So if M is small we have
M6⊂ sec N . Therefore M and N contain disjoint sets. �

Definition 3.6. A seminearness space (X,γ) satisfies (S) if it fulfills one (and
hence both) of the conditions formulated in proposition 3.5



Extensions of closure spaces 231

Remark that if (X,γ) satisfies (S) and M and N are different minimal small
stacks, M ∈M and N ∈N satisfying (ii) can be taken to be disjoint and open.
Hence in that case we have

∀x ∈ X : M 6∈ V(x) or N 6∈ V(x)

Next we generalize these ideas in order to introduce an even stronger sepa-
ration condition. Let (X,γ) be a seminearness space and let

Σ = {ẋ|x ∈ X}∪{M| minimal small nonconvergent}

Definition 3.7. Subsets D and B are said to be γ-disjoint if

∀P ∈ Σ : D 6∈ P or B 6∈ P

Clearly D and B are γ-disjoint if and only if

(i) D ∩B = ∅
(ii) for every M minimal small nonconvergent stack, D 6∈M or B 6∈M.

Definition 3.8. A seminearness space (X,γ) is said to satisfy (T) if minimal
small stacks M6= N contain sets M ∈M and N ∈N that are γ-disjoint.

Conditions (S) and (T) will play an important role in the investigation of
Hausdorff closure extensions. First we discuss the relation between the various
separation conditions.

Proposition 3.9.

(1) In a seminearness space we have

(T) ⇒ (S)
(2) In a concrete seminearness space we have

(T) ⇒ (S) ⇒ separated

Proof.

(1) Let M and N be (concentrated) minimal small, choose γ-disjoint sets
M ∈M and N ∈N . Then we have M ∩N = ∅.

(2) (X,γ) is concrete and satisfies (S). Let A be concentrated and let M
be a minimal small stack, M⊂A. Consider sec A which is small and
a minimal small stack N ⊂ sec A. It follows that M = N . Finally by
proposition 3.2 the space (X,γ) is separated.

�

From the proof of (2) we immediately have the following.

Corollary 3.10. If (X,γ) is concrete and satisfies (S) then every minimal
small concentrated stack M satisfies

M⊂ sec M

i.e. M is a linked system in the sense of [?].



232 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

No other implications between the conditions (T), (S) and ”separated” are
true in general (except those obtained by transitivity). We refer to section
?? for the summarizing diagrams. Example ?? provides a separated concrete
seminearness space which does not satisfy (S). Example ?? is a concrete semi-
nearness space satisfying (S) but not (T). Example 3 in [?] satisfies (T) but it
is not separated (and not concrete). Remark that this example moreover is a
nearness space.

In the case of a nearness space however some other implications become
true.

Proposition 3.11.
(1) For a nearness space we have

separated ⇒ (S) ⇔ (T)
(2) For a concrete nearness space we have

separated ⇔ (S) ⇔ (T)

Proof.
(1) Let (X,γ) be a separated nearness space. If M and N are minimal

small concentrated stacks then M and N are filters [?], [?]. If every
M ∈M would intersect every N ∈N then stack {M∩N|M ∈M,N ∈
N} would be a filter too and so it would be concentrated. Hence
M = N . So (X,γ) satisfies (S).

Next suppose (X,γ) is a nearness space satisfying (S). Let M and
N be minimal small. Since M and N are filters, both are concentrated
and so (S) implies that ∃M ∈ M,∃N ∈ N : M ∩N = ∅. Now every
minimal small P is a filter too, so M and N can not be both in P.

(2) Follows immediately from (1) in combination with proposition 3.9 (2).
�

That (S) 6⇒ separated, even in the nearness case follows from example 3 in
[?].

Next we discuss the impact of the separation conditions on extensions.

Proposition 3.12. A seminearness space is induced by a Hausdorff closure
extension if and only if it is separated, T1 and satisfies (S).

Proof. Suppose (Y,clY ) is a Hausdorff extension of (X,γ) then if M and N
are different minimal small stacks, by proposition 2.3 there are y and z in Y
such that M = VY (y)|X and N = VY (z)|X. It follows that y 6= z and then
disjoint sets can be chosen using the Hausdorff property of (Y,clY ). So (X,γ)
satisfies (S). That (X,γ) is separated and T1 follows from the fact that these
properties are hereditary.

Conversely, suppose (X,γ) is separated, T1 and satisfies (S). Consider

α = {M|M minimal small, concentrated, not convergent}
∪ {A|A small, openbased, not concentrated, not convergent}



Extensions of closure spaces 233

and construct the loose enlargement on

X′ = X ∪{yP|P ∈ α}

In view of paragraph 2.1, in order to conclude that (X′,cl′) is an exten-
sion, it suffices to prove that if D ∈ γ is openbased and not convergent in X
then stackX′D converges to some point in X′. Either D is concentrated and
then D ⊃ M for a unique minimal small concentrated stack M. In this case
stackX′D ⊃ stackX′M∩ ẏM for yM ∈ X′. Or D is not concentrated and then
stackX′D converges to yD ∈ X′.

Finally we prove that this loose extension is a Hausdorff closure space. Con-
sider two different points in X′. If each of them corresponds to a concentrated
minimal small stack, then (S) implies that disjoint neighborhoods can be found.
If at least one of the points corresponds to some openbased small stack A which
is not concentrated and not convergent, then sec A is not small and then the
argument developed in the proof of proposition 3.5 (ii) can be used to obtain
disjoint neighborhoods.

�

Proposition 3.13. A seminearness space (X,γ) is induced by a strict Haus-
dorff closure extension if and only if (X,γ) is concrete, T1 and satisfies (T).

Proof. Suppose (X,γ) is induced by a strict Hausdorff closure extension (Y,clY ).
Then by proposition 2.6 we already know that (X,γ) is T1 and concrete. In
paragraph 2.2 we remarked that (Y,clY ) is unique up to isomorphism. So we
have that (X̂, ĉl) is Hausdorff. In order to prove (T), let M 6= N be minimal
small stacks. By corollary 2.4 they correspond to different points y and z in X̂.
Consider disjoint basic neighborhoods O(A) and O(B) of y and z, respectively
where A,B ⊂ X. Then clearly int A ∈M and int B ∈N , so int A and int B
are γ-disjoint.

Conversely, suppose (X,γ) is T1, concrete and satisfies (T). Then we already
know that (X̂, ĉl) is a strict T1 closure extension. In order to prove that (X̂, ĉl)
is Hausdorff let y and z be different points. These points correspond to different
minimal small stacks in (X,γ) which therefore contain γ-disjoint sets A and
B. It follows that O(A) and O(B) are disjoint and belong to the respective
neighborhood collections VY (y) and VY (z). �

Remark that again our situation differs fundamentally from its topological
counterpart. The existence of a topological Hausdorff extension inducing (X,γ)
implies the existence of a strict topological Hausdorff extension [?]. Whereas
here for the existence of a Hausdorff closure extension only separated and (S)
are needed on (X,γ), and for the existence of a strict Hausdorff closure exten-
sion concreteness and (T) are involved and as announced in 3.10 these condi-
tions are not equivalent. In section ?? examples are listed showing that even
in the concrete case (S) plus separated does not imply (T).



234 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

4. Regularity and extensions.

Regularity was introduced in the context of prenearness spaces in [?]. So we
can apply the definition to our setting of seminearness spaces.

If A and B are subsets in (X,γ) one puts

B <µ A ⇐⇒ {A,X −B}∈ µ

where µ is the covering structure associated with γ. For a stack A one puts

A<µ = {A|∃B ∈A : B <µ A}

and for covers U and V one writes

V <µ U ⇐⇒ ∀V ∈V : ∃U ∈U : V <µ U

Using this notation one has the equivalence of the following statements
(i) ∀A∈ γ also A<µ ∈ γ
(ii) ∀U ∈ µ : ∃V ∈ µ : V <µ U

A seminearness space is said to be regular if it satisfies the previous equivalent
statements [?].

Next we introduce a stronger version of regularity by strengthening ”dis-
jointness” as we did before. Using γ-disjointness instead of disjointness we
again obtain equivalent statements.

Proposition 4.1. Let (X,γ) be a seminearness space and µ be the associated
covering structure. Let A and B be subsets of X. The following are equivalent:

(i) {A}∪{D|D and B γ-disjoint}∈ µ
(ii) ∀A small, if for every D ∈A the sets D and B are not γ-disjoint, then

A ∈A

We write B <<µ A if one and then both conditions stated in the previous
proposition hold. In view of the fact that

{A}∪{D|D and B γ-disjoint}≺{A,X −B}

we have that
B <<µ A ⇒ B <µ A

For a stack A let
A<<µ = {A|∃B ∈A : B <<µ A}

For covers U and V we write

V <<µ U ⇐⇒ ∀V ∈V : ∃U ∈U : V <<µ U

Using this notation we obtain the following equivalent statements. The proof
of the equivalence is quite similar to the equivalence based on <µ instead of
<<µ.

Proposition 4.2. The following are equivalent:
(i) ∀A∈ γ we have A<<µ ∈ γ
(ii) ∀U ∈ µ : ∃V ∈ µ : V <<µ U



Extensions of closure spaces 235

Definition 4.3. A seminearness space satisfies (R) if it fulfills one (and then
both) of the previous statements.

Clearly (R) implies regularity. Moreover as for nearness spaces every uniform
seminearness space is regular. However not every uniform seminearness space
satisfies (R), example ?? serves as a counterexample.

In general we have the following implications.

Proposition 4.4. If (X,γ) is a seminearness space then we have
(1) regular implies separated and regular implies (S)
(2) (R) implies regularity and (R) implies (T)

Proof.
(1) That a regular seminearness space is separated is proved analogously

to the nearness case. A regular seminearness space also satisfies (S). In-
deed: let M and N be small stacks and suppose ∀M ∈M,∀N ∈N : M ∩N 6= ∅.
Consider the stack M∩N and let U ∈ µ. Take V ∈ µ such that V <µ U.
Further let V ∈V∩M and let U ∈U be such that V <µ U. Now since
N is small we have

N ∩{U,X −V} 6= ∅

Clearly X − V 6∈ N and so finally we have U ∈ M∩N . So we can
conclude that M∩N is small and in case M and N are minimal small
this implies M = N .

(2) (R) implies (T). Let M and N be minimal small stacks. Suppose
M6= N and assume that M and N do not contain γ-disjoint sets. We
prove that M∩N is small. Let U ∈ µ and consider V ∈ µ such that
V <<µ U. Since M is small we can take V ∈V ∩M and then U ∈U
such that V <<µ U. Consider the uniform cover

W = {U}∪{D|D and V γ-disjoint}

Then N ∩W 6= ∅. Now by assumption on M and N , U must belong
to N . So finally U ∈ M∩N and the rest follows as in the previous
part.

�

In fact no other implications (except for those obtained by transitivity) hold.
Counterexamples for the nonvalid ones can be found in section ??.

Proposition 4.5. Let (X,γ) be a seminearness space. If M⊂ sec M for all
minimal small stacks M that do not converge, then we have

(i) D and B are γ-disjoint if and only if D and B are disjoint
(ii) B <<µ A if and only if B <µ A
(iii) (X,γ) has (R) if and only if it is regular

Proof. We only need to show (i). Clearly if M ⊂ sec M then disjoint sets D
and B can not both belong to M �



236 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

In every nearness space minimal small stacks are filters and so the statements
in the previous proposition hold, in particular for nearness spaces, regularity
is equivalent to (R). Applying corollary 3.10 we obtain that in every concrete
seminearness space in which all minimal small stacks are concentrated and in
which (S) holds, also M⊂ secM holds for every minimal small stack. It follows
that every regular concrete seminearness space in which all minimal small stacks
are concentrated satisfies (R). In particular a closure (seminearness) space is
regular if and only if it satisfies (R). It can be easily seen that analogously to
the topological case, a closure (seminearness) space X is regular if and only if

∀x ∈ X : V(x) has a closed base

However even on a nearness space the condition that ”clγA is small whenever
A is small” (called weakly regular in [?]) is strictly weaker than regularity.

Next we investigate the regularity of the strict extension of a concrete semi-
nearness space.

Proposition 4.6. Let (X,γ) be a concrete T1 seminearness space, then it is
induced by a strict regular extension if and only if (X,γ) satisfies (R).

Proof. Suppose (X,γ) satisfies (R). We prove that the strict T1 extension
(X̂, ĉl) is regular. Let yM ∈ X̂ where M is a minimal small stack that is
not convergent (or alternatively let x ∈ X). Let O(A) be a basic open set
containing yM (or x) where A is some subset of X. So A ∈M (or A ∈V(x)).
Since M<<µ = M (V(x)<<µ = V(x)) we have A ∈M<<µ (A ∈V(x)<<µ) and
so we can find B ∈M (B ∈V(x)) such that B <<µ A. Now we prove that

ĉl O(B) ⊂ O(A)

Let yN ∈ ĉl O(B) where N is minimal small, not convergent. Then for every
D ∈N we have

O(D) ∩O(B) 6= ∅
It follows that for every D ∈ N the sets D and B are not γ-disjoint. Hence
A ∈N and then yN ∈ O(A).

Moreover if x ∈ ĉl O(B) then for every D ∈V(x) we have O(D)∩O(B) 6= ∅.
Again it follows that for every D ∈ V(x) the sets D and B are not γ-disjoint
and therefore A ∈V(x) and x ∈ O(A).

Conversely suppose that (X,γ) is induced by a strict regular T1 extension.
By proposition 2.6 this means that (X̂, ĉl) is regular. We prove that (X,γ)
satisfies (R).

Let A be small. Since (X,γ) is concrete we can take a minimal small stack
M such that M⊂A. We prove that M⊂M<<µ.

Let A ∈M. If M is not convergent we have yM ∈ O(A) (or if M = V(x) we
have x ∈ O(A)). Since O(A) is open in (X̂, ĉl) it contains a closed neighborhood
and so we can find B ⊂ X such that yM ∈ O(B) (or x ∈ O(B)) and such that

O(B) ⊂ ĉl O(B) ⊂ O(A)



Extensions of closure spaces 237

But then we have B ∈M (or B ∈V(x)) and it is clear that B can be assumed
to be open. Now B <<µ A. Indeed let D be small such that for every D ∈D
the sets D and B are not γ-disjoint. Let N be minimal small with N ⊂ D.
Then also for every N ∈ N the sets N and B (which can be considered to be
open) are not γ-disjoint. It follows that

O(N) ∩O(B) 6= ∅,∀N ∈N

and finally that yN ∈ ĉl O(B) if N is not convergent, or that x ∈ ĉl O(B)
if N = V(x). So we have yN ∈ O(A) or x ∈ O(A), respectively and we can
conclude that A ∈N . �

It is known that for a topological extension, regularity of the extension
implies strictness. In our setting of closure extensions this is no longer true. A
seminearness space can be induced by a regular T1 extension without having a
strict regular T1 extension. In examples ?? and ?? we’ll prove that the structure
is induced by a regular T1 extension. However neither ?? nor ?? satisfies (R).

5. Summarizing diagrams and examples.

5.1. For closure spaces and for concrete nearness spaces we have that several
notions coincide, namely that ”T ⇐⇒ S ⇐⇒ separated” and that ”R ⇐⇒ regularity”.

5.2. For arbitrary nearness spaces we have the implications of figure ??.

R

Separated

T

Regular S

Uniform

Figure 1. Implications for nearness spaces

The only valid implications are those indicated and those obtained by tran-
sitivity.

5.3. As an example of a nonconcrete nearness space satisfying (S) but that
is not separated, one can consider example 3 in [?]. That this example indeed
satisfies condition (S) follows from the fact that the only concentrated minimal
small stacks are the pointfilters.



238 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

R

Regular

Separated

T

S

Uniform
5.7

5.7
5.8

5.6 5.6

5.8

5.8
5.7

Figure 2. Implications for concrete seminearness spaces

5.4. For concrete seminearness spaces we proved the implications in figure ??.
Again no other implications hold except for those obtained by transitivity.

The numbers refer to the examples presented below and these are counterex-
amples for the reversed arrows.

5.5. For general seminearness spaces the diagram is as in figure ??.

R

Separated

T

Regular S

Uniform

Figure 3. Implications for seminearness spaces

As a counterexample showing that (S) does not imply separated in the non-
concrete case, we again refer to example 3 in [?].

5.6. A concrete and separated T1 seminearness space which does not satisfy
(S). Therefore it is nonregular, nonuniform and satisfies neither (R) nor (T).

Let X = R2 and define

M = stack {pr−11 (c)|c ∈ R}
N = stack {pr−12 (c)|c ∈ R}



Extensions of closure spaces 239

For a stack A we define

A∈ γ ⇐⇒ M⊂A or N ⊂A or ẋ ⊂A for some x ∈ X

The underlying closure is the discrete one. Clearly every small stack contains
a unique minimal small stack, but M and N do not contain disjoint sets and
so (X,γ) does not satisfy (S).

5.7. A concrete T1 seminearness space which is uniform (and even zerodi-
mensional) and so it is regular and satisfies (S) and is separated. However it
satisfies neither (T) nor (R).

Let A,B,C be three pairwise disjoint sets with more than one point and
X = A∪B ∪C. In order to define γ on X consider the following stacks

M = stack {A,B}

N = stack {B,C}

P = stack {A,C}

Define a stack A to be small if and only if

M⊂A or N ⊂A or P ⊂A or ẋ ⊂A for some x ∈ X

Then the pointfilters ẋ for x ∈ X are the only concentrated minimal stacks,
and the other minimal stacks M,N ,P are not concentrated.

(X,γ) does not satisfy (T) since for instance for M and N neither of the
disjoint sets A and B, A and C or B and C are γ-disjoint. It follows that (X,γ)
does not satisfy (R).

However (X,γ) is uniform since its collection µ of uniform covers is generated
by the following collection µ′ = {U1,U2,U3} of partitions

U1 = {A,B}∪{{x}|x ∈ C}

U2 = {B,C}∪{{x}|x ∈ A}

U3 = {A,C}∪{{x}|x ∈ B}

It follows that (X,γ) is regular, also separated and satisfies (S). So the strict
extension (X̂, ĉl) is a T1 extension that is not Hausdorff and not regular. Re-
mark however that the loose extension constructed by adding different points
for the minimal small stacks that do not converge, is regular and T1.

5.8. A concrete T1 seminearness space which is uniform (and even zerodimen-
sional) and therefore is regular. It satisfies (T) and hence also (S) and it is
separated. However it does not satisfy (R).

Let A,A′,P,P ′,Q and Q′ be pairwise disjoint sets with more than one point
and let

X = A∪A′ ∪P ∪P ′ ∪Q∪Q′



240 D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

In order to define γ on X consider the following stacks

A = stack {A,A′}
P = stack {P,P ′,A}
Q = stack {Q,Q′,A}
B = stack {P,Q}

Define a stack S to be small if and only if

A⊂S,P ⊂S,Q⊂S,B ⊂S or ẋ ⊂S for some x ∈ X

Clearly (X,γ) is concrete and T1. Use the fact that A′,P ′,Q′ are sets belong-
ing to just one nonconvergent minimal small stack to see that (X,γ) satisfies
(T).

Again (X,γ) is uniform and in fact (X,µ) is generated by a collection of
partitions. So (X,γ) is regular.

However (R) is not satisfied. Let’s concentrate on A and consider A ∈ A.
The sets P and A are not γ-disjoint since they both belong to P. Also Q and
A are not γ-disjoint. It follows that for

U = {A}∪{D|D and A γ-disjoint}

we have U∩B = ∅. So U 6∈ µ and therefore A <6<µ A. Clearly this implies that
A<<µ 6∈ γ. It follows that the strict extension (X̂, ĉl) of (X,γ) is a Hausdorff
closure space that is not regular. Remark however that the loose extension
constructed by adding different points for the minimal small stacks that do not
converge, is a regular T1 extension.

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Received November 2001

Revised October 2002

D. Deses, A. De Groot-Van der Voorde, E. Lowen-Colebunders

Department of Mathematics, Vrije Universiteit Brussel, 1050 Brussels, Bel-
gium

E-mail address : diddesen@vub.ac.be, avdvoord@vub.ac.be, evacoleb@vub.ac.be