@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 391–419

Hyperconvergences

Szymon Dolecki and Frédéric Mynard

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

Abstract. The hyperconvergence (upper Kuratowski convergence)
is the coarsest convergence on the set of closed subsets of a convergence
space that makes the canonical evaluation continuous. Sundry reflective
and coreflective properties of hyperconvergences are characterized in
terms of the underlying convergence.

1. Introduction.

Convergences on hyperspaces (spaces of closed sets) have been attracting
attention due to their numerous applications, but also for theoretic reasons.
Closed sets (of a convergence) can be identified with continuous maps (from
that convergence) valued in the Sierpiński topology $ = {∅,{1},{0, 1}}. There-
fore, a hyperspace is an instance of a function space.

A convergence on a hyperspace that is of particular interest, is the coars-
est among those convergences for which the natural evaluation map (from the
product of a convergence space with its hyperspace) valued in the Sierpiński
space is continuous. We shall call it the hyperconvergence (or the upper Ku-
ratowski convergence). It follows from the definition that hyperconvergences
constitute a particular instance of continuous convergences ; recall that the con-
tinuous convergence (of a convergence) is the coarsest convergence on the set of
continuous maps from that convergence (valued in another convergence space)
for which the natural evaluation map is continuous.

On the other hand, hyperconvergences play a universal role among all con-
tinuous convergences (valued in topologies). This is because the Sierpiński
topology is initially dense in the category of all topologies, which means that
every topology is the initial object with respect to maps valued in the Sierpiński
space [1]. Therefore, every continuous convergence valued in a topology is the
initial object with respect to maps valued in hyperconvergences. This fact
has important consequences for continuous convergences as well as for quotient
maps.



392 Szymon Dolecki and Frédéric Mynard

Hyperconvergence can be characterized as the (lower) order convergence on
the complete lattice of closed sets; transposed onto the lattice of open sets, it
is the Scott convergence ([13] in the case of open sets of a topological space).

We study hyperspaces with respect to convergences that are not necessarily
T1. This is not because of a desire of utmost generality, but because of the very
nature of the considered topic. Indeed the hyperconvergence (for a nonempty
convergence) is never T1, but is always T0. Another peculiar property of hy-
perconvergences is hypercompactness : every filter converges.

To express properties of hyperconvergences in terms of the corresponding
properties of the underlying convergences, is an essential element of our quest.
Because hyperconvergences are defined on the spaces of closed sets, hypercon-
vergence properties typically correspond to hereditary properties (with respect
to open sets) of the underlying convergences. For example, the hyperconver-
gence of a topology is topological if and only if the underlying topology is
core-compact [13]. The very fact that the hyperconvergence of a topology need
not be topological has been at the origin of convergence theory [5].

In this survey paper we not only present a collection of our recent results
related to hyperconvergence [11], [17], [16], [15], [14], but also refine several of
them and clarify certain links between them. Therefore we have tried to make
this paper self-contained providing full proofs in most cases.

2. Notation.

Following Engelking, A ⊂ X means that A is a (not necessarily proper)
subset of X, and Ac = X\A when X has been fixed. We consider now families
of subsets of a fixed set. A family B is finer than a family A (in symbols,
A≤B) if for every A ∈A there exists B ∈B such that B ⊂ A. Families of sets
E and H mesh (in symbols, E#H) provided E ∩ H 6= ∅ for each E ∈ E and
every H ∈H; we denote by E# the grill of E, that is, the family of all sets that
intersect every element of E. The symbol Ec stands for the family {Ec : E ∈E}.
If F is a filter, then we denote by βF the set of all the ultrafilters which are
finer than F.

If a : 2Y → 2Y and if A is a family of subsets of Y , then we denote by
a\A = {aA : A ∈A}

the a-regularization of A. If now H is a class of families of sets, then a\H =
{a\H : H∈ H}.

On the other hand, if V(y) is a family of subsets of X for every y ∈ Y , then
for a family A of subsets of Y ,

V(A) =
⋃
A∈A

⋂
y∈A
V(y)

is the contour of V(·) along A. If A = {A} then we abridge V(A) = V({A}).
If now X = Y and {V(x) : x ∈ X} is given and we define x ∈ aVA if and only
if A ∈V#(x), then we have

(2.1) V(A)#B ⇐⇒A#a\VB.



Hyperconvergences 393

By Oξ(x) we denote the family of all ξ-open sets which contain x. Therefore,
Oξ(A) stands for the family of all ξ-open sets which include A and Oξ(A) =⋃
A∈AOξ(A).
Finally, we denote respectively by f−(y),f−(B) and f−(B) the preimage

by f of y,B and B. If H ⊂ X ×Y and A ⊂ X then the image of A by H is
denoted by HA while H−B stands for the preimage of B ⊂ Y by H.

3. Hyperspace.

The hyperspace associated with a topology (more generally, with a conver-
gence) τ is the set of all τ-closed subsets. It is well known that there exists a
one-to-one correspondence between closed subsets and continuous maps which
take values in a two-element set endowed with the Sierpiński topology $. In-
deed, if τ is a convergence (see Section 5) on a set X and A ⊂ X, then the
indicator map

(3.1) 〈x,A〉 =
{

0, if x ∈ A
1, if x /∈ A

is continuous (in the first variable) from τ to $ = {∅,{1},{0, 1}} if and only if A
is τ-closed. Therefore C(τ, $) stands for the set of all τ-closed sets, equivalently
for the set of all maps continuous from τ to $. In particular, the constant maps
〈x,∅〉 = 1 and 〈x,X〉 = 0 correspond to the empty set and the whole of X
respectively.

We shall sometimes call hyperpoints the elements of a hyperspace, hypersets
the subsets of a hyperspace (that is, families of closed sets), and hyperfilters
the filters on a hyperspace.

The Sierpiński topology plays an exceptional role among other topologies:
it is initially dense in the category of all topologies. This means that every
topology τ is the initial object with respect to maps valued in the Sierpiński
topological space:

(3.2) τ =
∨

f∈C(τ,$)

f−$,

where f−θ denotes the initial convergence of θ by f. Indeed, the statement
above is an immediate consequence of the representation of closed sets with the
aid of the coupling map (3.1) continuous in the first variable for every fixed A.

4. Polarities, Galois connections.

We will see later that minimality of hyperconvergence can be rephrased in
terms of polarity. A map f : X → Y (between complete lattices X,Y ) is called
an inverse polarity whenever

f(
∧
A) =

∨
f(A)



394 Szymon Dolecki and Frédéric Mynard

for every A ⊂ X. Each inverse polarity admits its adjoint map f∗(y) =
∧
{x :

y ≥ f(x)} which is also a polarity. A couple of maps

(4.1) f : X � Y : g

is called a (−,−)-connection if f and g are order-inverting and if gf and fg are
contractive, that is, gf(x) ≤ x and fg(y) ≤ y. It is well known [3] that (4.1) is
a (−,−)-connection if and only if f is an inverse polarity and g = f∗. It is often
handy to invert order on one or both the lattices X,Y . If we invert the order
on X only, then we obtain a (+,−)-connection, with f and g order-preserving,
with gf expansive (that is, gf(x) ≥ x) and fg contractive. Similarly we define
(−, +)-connections and (+, +)-connections, and refer to all of them as Galois
connections.

A projector is an order-preserving, contractive and idempotent map, while
an order-preserving, expansive, idempotent map is called a coprojector. An
important consequence of the definitions above is that if (4.1) is a (−,−)-
connection, then f = fgf and gfg = g, hence gf and fg are projectors.
Moreover, the restrictions

f : fix(gf) � fix(fg) : g

constitute a couple of reciprocal lattice inverse isomorphisms. Of course, if
(4.1) is a (+,−)-connection, then gf is a coprojector while fg is a projector.
Finally, let us mention that (4.1) is a (−,−)-connection if and only if f(x) ≥ y
is equivalent to x ≤ g(y) for every x and y.

5. Convergence.

We refer to [11], [17] for details of convergence theory. A convergence ξ on
a set X is a relation between filters F on X and elements x of X

x ∈ limξF

such that limξF ⊂ limξG if F ≤ G and x ∈ limξ{x}↑ for each x ∈ X, where
{x}↑ stands for the principal ultrafilter of x; the latter condition is that of
strictness. Please, have in mind that various authors give various (slightly
different) definitions of convergence; a most frequent additional axiom is the
one that we use to define prototopologies (see Section 11).

The adherence of a family H of sets is defined by

adhξH =
⋃
E#H

limξE.

If ξ is a convergence then |ξ| stands for the set on which ξ is defined. A
convergence ζ is finer then a convergence ξ (ξ is coarser than ζ) whenever
limζF ⊂ limξF for every filter F. If ξ is a convergence on X and τ is a con-
vergence on Y, then a map f : X → Y is continuous if f(limξF) ⊂ limτf(F).
The final convergence (the finest convergence on Y for which f is continuous
from ξ) is denoted by fξ and the initial convergence (the coarsest convergence
on X for which f is continuous to τ) is denoted by f−τ.



Hyperconvergences 395

A set V is a ξ-vicinity of x if V ∈F for every filter F such that x ∈ limξF;
the family Vξ(x) of vicinities of x is a filter. A convergence ξ is a pretopology
if and only if x ∈ limξVξ(x) for every x. A set O is ξ-open if O ∈ F for every
filter F such that limξF ∩ O 6= ∅ (ξ-closed if its complement is ξ-open). In
other words, a set is ξ-open if and only if it is a ξ-vicinity of each its elements.
A set W is a ξ-neighborhood of x if it includes a ξ-open set that contains x;
the set of all ξ-neighborhoods of x is denoted by Nξ(x). A convergence ξ is
a topology if and only if x ∈ limξNξ(x) for every x. Of course, a topological
closure aNξ = clξ is defined via (2.1). It is the closure for a topology Tξ called
topologization of ξ.

A family G of filters is a base for a convergence ξ whenever x ∈ limξF implies
the existence of G ∈ G such that G ≤F and x ∈ limξG.

6. Hyperconvergence.

The hyperconvergence (also called the upper Kuratowski convergence) is the
power convergence with respect to the Sierpiński topology. In other words,
if ξ is a convergence on X, then the hyperconvergence [ξ, $] associated with
ξ is the coarsest convergence on C(ξ, $) such that 〈·, ·〉 : ξ × C(ξ, $) → $ is
continuous. We say that ξ is a primal convergence of [ξ, $] and that [ξ, $]
is the dual convergence of ξ. It follows directly from the definition of power
convergence that A0 ∈ lim[ξ,$]G if and only if for every x ∈ |ξ| and each filter
F on |ξ| such that x ∈ limξF,

(6.1) 〈x,A0〉 ∈ lim$〈F, G〉.

Here G is a hyperfilter and A0 is a hyperpoint. The filter 〈F, G〉 is generated
by 〈F,G〉 = {〈x,G〉 : x ∈ F,G ∈G}.

It is possible to characterize the hyperconvergence in terms of the primal
convergence. To this end we will need a notion of reduced filter. Let us have
in mind that if G is a ξ-hyperfilter (that is, a filter on C(ξ, $)) then every
G ∈ G is a ξ-hyperset (that is, a family of ξ-closed sets). The reduced filter of
a hyperfilter G is defined in [10] by

rG ≈{
⋃
A∈G

A : G ∈ G},

where F ≈B means that F is generated by B. Notice that the non degeneracy
of a hyperfilter G does not imply the non degeneracy of the reduced filter rG;
indeed, rG is degenerate if and only if there exists ∅ 6= G ∈ G such that⋃
A∈G A = ∅, that is, whenever G = {∅} which means that G = {∅}

↑ (the
principal ultrafilter of the empty set ∅). Of course, lim[ξ,$]{∅}↑ contains the
hyperpoint ∅ and thus every ξ-closed subset A of |ξ|. On the other hand,
adhξr({∅}↑) = adhξ∅↑ = ∅.

Proposition 6.1. [10] Let G be a hyperfilter on C(ξ, $). Then

(6.2) A0 ∈ lim[ξ,$]G ⇐⇒ adhξrG ⊂ A0.



396 Szymon Dolecki and Frédéric Mynard

Proof. We have already noted that A0 ∈ lim[ξ,$]G is equivalent to (6.1) for
every x ∈ |ξ| and each filter F on |ξ| such that x ∈ limξF. Because the only
$-neighborhood of 0 is the whole of {0, 1}, the formula above is significant only
in the case of 〈x,A0〉 = 1, that is, x /∈ A0. Therefore (6.1) means that if x /∈ A0
then there exist F ∈F and G ∈ G such that F ∩

⋃
A∈G A = ∅, that is, F does

not mesh with rG. We conclude that if x /∈ A0, then x /∈ adhξrG, and thus the
proof is complete. �

Corollary 6.2. A hyperfilter G converges to A0 in [ξ, $] if and only if for every
filter F such that limξF \A0 6= ∅, there is G ∈ G for which

⋂
A∈G A

c ∈F.

Proof. By (6.2) A0 ∈ lim[ξ,$]G if and only if every filter F which meshes with
rG does not ξ-converge to an element of Ac0. Therefore limξF∩Ac0 6= ∅ implies
that F does not mesh with rG, that is, there exist G ∈ G and F ∈F such that
F ∩

⋃
A∈G A = ∅, in other words,

⋂
A∈G A

c ∈F. �

A convergence is hypercompact provided that every filter converges. It fol-
lows from (6.2) that

Proposition 6.3. Each hyperconvergence is hypercompact.

Proof. Indeed |ξ| ∈ lim[ξ,$]G for every filter G, because adhξrG ⊂ |ξ|. �

A convergence is said to be T1 if all its singletons are closed. Except for
the trivial case (of the discrete topology on a singleton), no hypercompact
convergence is T1; in fact, if τ is a hypercompact convergence on Y and y0 6= y1,
then ∅ 6= limτ{y0,y1}↑ ⊂ limτ{y0}↑ ∩ limτ{y1}↑. Hence if τ were T1, then
limτ{y0}↑ ∩ limτ{y1}↑ = ∅.

Proposition 6.4. If |ξ| is non empty, then the hyperconvergence [ξ, $] is not
T1.

More precisely, if A ∈ C(ξ, $), then

(6.3) cl[ξ,$]{A} = {B ∈ C(ξ, $) : A ⊂ B},

so that |ξ| ∈ cl[ξ,$]{∅}. Therefore, if X 6= ∅ and ξ is any convergence on X,
then the ξ-closed sets ∅ and X are distinct and X ∈ cl[ξ,$]{∅}.

Proposition 6.5. Each hyperconvergence is T0.

Proof. Indeed, if A0,A1 ∈ C(ξ, $) and x ∈ A1 \ A0, then by (6.3) x ∈ A for
every A ∈ cl[ξ,$]{A1}, hence A0 /∈ cl[ξ,$]{A1}. �

The hyperconvergence of the (unique) convergence on a singleton (this is
necessarily the discrete topology ι) is homeomorphic to the Sierpiński topology.
In fact the (ι-closed) subsets of {x} are ∅ and {x}; for [ι, $], the principal
hyperfilter of ∅ converges to ∅ and to {x}, while the principal hyperfilter of
{x} converges to {x}; hence the map which associates {x} to 0 and ∅ to 1 is
a homeomorphism.



Hyperconvergences 397

7. Point and star topologies.

With a view to characterizing reduced filters, we associate with each con-
vergence two special topologies [4], [11]. If ξ is a convergence on X, then
clξ•x = clξ{x} defines a (binary) relation on X. For our purposes, it is handy
to use a special notation for the inverse relation, namely,

clξ∗y = cl
−
ξ•y = {x : y ∈ clξ{x}}.

As usual, the image and the preimage of sets by a relation are

clξ•A =
⋃
x∈A

clξ{x} and clξ∗B = {x : clξ{x}∩B 6= ∅} =
⋃
y∈B

clξ∗y.

Both the operations clξ• and clξ∗ turn out to be topological closures. In
fact, it is straightforward that they are expansive and that clξ•∅ = clξ∗∅ = ∅.
The idempotency of clξ• and of clξ∗ follows immediately from that of clξ. As
for finite additivity, we get more than needed: as the operations are relations,
they are (fully) additive:

clξ•(
⋃
i∈I

Ai) =
⋃
i∈I

clξ•Ai; clξ∗(
⋃
i∈I

Ai) =
⋃
i∈I

clξ∗Ai.

The topology ξ• is called the point topology of ξ and the topology ξ∗ is called
the star topology of ξ. If ξ is T1, then ξ• and ξ∗ are equal to the discrete
topology. Of course, ξ• ≥ ξ, but this is not necessarily the case with ξ∗.

Example 7.1. Consider the discrete topology ι on {0, 1}. Then the hyper-
convergence [ι, $] is a topology and for instance cl[ι,$]•{{0}} = cl[ι,$]{{0}} =
{{0},{0, 1}} while cl[ι,$]∗{{0}} = {∅,{0}}.

As both the point- and the star- closures are additive, the collection of all
ξ•-closed (respectively, ξ∗-closed) sets is also that of open sets of some topology.

Proposition 7.2. For every convergence ξ, a set is ξ•-closed if and only if it
is ξ∗-open.

Proof. If y /∈ clξ•A = A then x /∈ clξ∗y for every x ∈ A, that is, A∩ clξ∗y = ∅.
In other words, clξ∗Ac ⊂ Ac. �

The topology ξ• has been defined with the aid of the ξ-closures of single-
tons. Because there exists a unique filter that contains the singleton {x},
namely the principal ultrafilter {x}↑, clξ•x = clξ{x} = limTξ{x}↑. Therefore
y ∈ limξ•

∧
x∈clξ∗y

{x}↑ = limξ•{clξ∗y}↑ and thus clξ∗y is a neighborhood of y
with respect to ξ•. Actually, another way of proving that a set is ξ•-open if
and only if it is ξ∗-closed is to notice that

Lemma 7.3. Nξ•(x) is the principal filter generated by clξ∗x.

Proof. Observe that W ∈ Nξ•(x) whenever x ∈ limξ•{y}↑ = clξ•y implies
that y ∈ W , equivalently clξ∗x ⊂ W. This means that clξ∗x is the least ξ•-
neighborhood of x. �



398 Szymon Dolecki and Frédéric Mynard

Therefore, by (2.1), for two families A and B,

(7.1) cl\ξ∗A#B ⇐⇒A#cl
\
ξ•B.

8. Reduced and erected filters.

If ξ is a convergence on X, then for each subset H of X, we define the erected
set eξH = {A = clξA : A ⊂ H}, and for a filter H on X, the erected filter of H

(8.1) e\ξH = {eξH : H ∈H}.

The erected filter of a non degenerate filter is never degenerate. Indeed, {∅}
belongs to every element of an erected filter.

The (possibly degenerate) filter re\ξH is generated by the sets of the form
intξ∗H =

⋃
clξA=A⊂H A, where H ∈H, that is, int

\
ξ∗H = re

\
ξH, and thus

re\ξH≥H.
A hyperset G is ξ-stable if B = clξB ⊂ G ∈ G implies B ∈ G. Notice

that G is ξ-stable if and only if Gc = Oξ(Gc). A hyperset G is ξ-saturated if
B = clξB ⊂

⋃
A∈G A implies that B ∈ G. A hyperfilter G on C(ξ, $) is ξ-

stable (respectively, ξ-saturated ) if it admits a filter base consisting of ξ-stable
(respectively, ξ-saturated) hypersets. It is straightforward that

Proposition 8.1. A hyperfilter is an erected filter if and only if it is saturated.

Proposition 8.2. Let F be a ξ-reduced filter and let G be a stable ξ-hyperfilter.
Then F meshes with rG if and only if the filters G and e\ξF restricted to C(ξ, $)\
{∅} mesh.

Proof. By definition, a ξ-reduced filter F meshes with rG if and only if
⋃

clξA=A⊂F
A ∩

⋃
B∈G B 6= ∅ for every F ∈ F and G ∈ G. Equivalently, there is a non

empty ξ-closed set A ∈ eξF ∩G for every F ∈ F and G ∈ G, because G is
stable. �

If now G is an arbitrary hyperfilter on C(ξ, $), then e\ξrG is coarser than
G. Indeed, a base of rG consists of

⋃
A∈G A where G ∈ G; therefore, e

\
ξrG

admits a base of the form {B = clξB ⊂
⋃
A∈G A} with G ∈ G, and since

G ⊂{B = clξB ⊂
⋃
A∈G A},

e\ξrG ≤ G.
It follows that

Proposition 8.3. The hyperconvergence [ξ, $] has a convergence base consist-
ing of saturated hyperfilters.

Proof. If A ∈ lim[ξ,$]G, then adhξr(e
\
ξrG) = adhξrG ⊂ A, hence A ∈ lim[ξ,$]e

\
ξrG;

as e\ξrG is erected, it is saturated. �

Proposition 8.4. [16] A filter is a reduced filter if and only if it is regular for
the point topology if and only if it is open for the star topology.



Hyperconvergences 399

Proof. If G ∈ G, and if x ∈
⋃
A∈G A, then there is A ∈ G and thus clξ{x} ⊂

clξA = A ⊂
⋃
A∈G A, so that rG is regular for the point topology of ξ. Con-

versely, if clξ•H ⊂ H, then
⋃

clξA=A⊂H A ⊂
⋃
x∈H clξ{x} = clξ•H ⊂ H, that

is, if H is regular for ξ•, then re\ξH = H. By definition, a filter is ξ
•-regular

if and only if it admits a base of ξ•-closed sets, that is, by Proposition 7.2, of
ξ∗-open sets; equivalently the filter is ξ∗-open. �

In general
int\ξ∗H = re

\
ξH≥H≥ cl

\
ξ•H.

By the proposition above, H = rG for some hyperfilter G on C(ξ, $) if and only
if int\ξ∗H = H = cl

\
ξ•H. Consequently, we shall use the term ξ-reduced filter

for regular for the point topology of ξ. Let us denote by cl\ξ•S the set of filters
that are regular for the point topology of ξ.

If we observe that G0 ≤ G1 implies r G0 ≤ rG1 and H0 ≤ H1 implies
e\ξH0 ≤ e

\
ξH1, then we have proved that

Theorem 8.5. The operations of erection e\ξ and of reduction r constitute
a Galois (+,−)-connection. Hence they form a lattice isomorphism between
reduced filters and filters based in saturated families.

It follows that for every family {Hi : i ∈ I} of filters on |ξ|,

(8.2) e\ξ

(∨
i∈I
Hi

)
=
∨
i∈I

e\ξ Hi,

and for every family {Gi : i ∈ I} of filters on C(ξ, $),

(8.3) r(
∧
i∈I

Gi) =
∧
i∈I

rGi.

Another immediate consequence of general properties of Galois connections, is
that

(8.4) H≤ rG ⇐⇒ e\ξH≤ G.

Because [ξ, $] has a convergence base consisting of ξ-saturated filters,

(8.5) adh[ξ,$]G = adh[ξ,$]e
\
ξr G.

9. Reflective and coreflective properties of convergences.

The category of convergences with continuous maps as morphisms is concrete
over the category of sets. This means that every morphism from ξ to τ is
uniquely determined by a map from |ξ| to |τ|. A (covariant) functor F is a
map on morphisms which preserves the composition; as every object ξ can be
identified with the identity morphism ιξ : ξ → ξ, each functor F maps also
objects following the rule ιF(ξ) = F(ιξ). A functor F between two concrete
categories (over the category of sets) is concrete whenever each morphism and
its image by F have the same underlying map.



400 Szymon Dolecki and Frédéric Mynard

A concrete subcategory of convergences is reflective (respectively, coreflec-
tive) if it is stable for initial (respectively, final) convergences; in other words,
M is reflective if τ ∈ M implies f−τ ∈ M, and

∨
i∈I τi ∈ M whenever τi ∈ M

for each i ∈ I (M is coreflective if ξ ∈ M implies fξ ∈ M, and
∧
i∈I ξi ∈ M

whenever ξi ∈ M for each i ∈ I).
Every concrete functor in the category of convergences is determined by its

action on objects [1, Remark 5.3]. An order-preserving map M on convergences
such that |Mξ| = |ξ|, is a concrete functor if and only if M(f−τ) ≥ f−(Mτ)
(equivalently, f(Mξ) ≥ M(fξ)) for every f and every τ on the domain (respec-
tively, each ξ on the range) of f. A concrete functor is called a reflector if it is
a projector, a coreflector if it is a coprojector.

From our point of view, a property of convergences is tantamount to a sub-
category of convergences. We shall consider reflective properties, like topologic-
ity, pretopologicity (corresponding to reflective subcategories) and coreflective
properties, like countable character, Fréchetness, sequentiality (corresponding
to coreflective subcategories). A reflective concrete subcategory can be de-
scribed as the class of those convergences τ for which

(9.1) Mτ ≥ τ,

where M is a concrete functor; a coreflective concrete subcategory is the class
of those convergences τ for which

(9.2) τ ≥ Mτ,

where M is a concrete functor.
If J is a class of filters possibly depending on convergence, then a convergence

ξ is called adherence-determined by J if limξF =
⋂

J(ξ)3H#F
adhξH for every filter

F [7]. Let

(9.3) limAJξF =
⋂

J(ξ)3H#F

adhξH.

We assume that ζ ≤ ξ implies J(ζ) ⊂ J(ξ), J(AJξ) = J(ξ) and that G ∈ J(ξ)
implies that (the filter generated by) f−(G) belongs to J(ξ). Then AJ is a
reflector. If J does not depend on convergence, then the first two conditions
are automatically fulfilled. A convergence is a pseudotopology, paratopology,
pretopology if it is adherence-determined by the class S (of all filters), Pω (of
countably based filters), P (of principal filters), respectively. These classes of
filters are independent of convergence; moreover the preimage by a map of
a filter from any of these classes, remains in the class. The corresponding
reflectors S,Pω,P are called the pseudotopologizer, the paratopologizer and the
pretopologizer. A convergence is a topology if it is adherence-determined by the
class T of principal filters of closed sets. The corresponding reflector T is the
topologizer.



Hyperconvergences 401

A convergence ξ is called E-based (or, based in E) if x ∈ limξF implies the
existence of a filter G ∈ E such that G ≤F and x ∈ limξG [7]. Let

(9.4) limBEξF =
⋃

E(ξ)3G≤F

limξG.

We assume that ζ ≤ ξ implies E(ζ) ⊂ E(ξ) and E(BEξ) = E(ξ) and that
G ∈ E implies that (the filter generated by) f(G) belongs to E. Then BE is a
coreflector. If E does not depend on convergence, then the first two conditions
are automatically fulfilled. For instance, a convergence is first-countable (or
of countable character ) if and only if it is based in the class Pω (of countably
based filters); we denote by First = BPω the corresponding coreflector.

We shall also consider the categories of convergences τ characterized by the
inequality

τ ≥ JEτ,
where J is a concrete reflector and E is a concrete coreflector. Such categories
are coreflective, by (9.2). In particular, τ is bisequential if τ ≥ SFirstτ, strongly
Fréchet if τ ≥ PωFirstτ, Fréchet if τ ≥ PFirstτ and sequential if τ ≥ TFirstτ.

10. Power convergence.

The power convergence (or the continuous convergence) [ξ,σ] of ξ with re-
spect to σ is the coarsest convergence on the set C(ξ,σ) of continuous maps
from ξ to σ for which the natural evaluation map is continuous. This is the only
convergence structure satisfying the exponential law for arbitrary convergences
ξ,τ,σ:

(10.1) [ξ × τ,σ] = [τ, [ξ,σ]],
where = stands for a homeomorphism via the transposition map tf(y)(x) =
f(x,y). Of course, hyperconvergences constitute a special case of power con-
vergences, namely with respect to σ = $.

Recall that for a map f : X → Y , the map f∗ : ZY → ZX is defined by
f∗(h) = h ◦ f. It turns out that if f : ξ → τ is a continuous map, then (the
restriction of ) f∗ : [τ,σ] → [ξ,σ] is also continuous. In particular, if ξ ≥ τ then
C(τ,σ) ⊂ C(ξ,σ) and the injection from [τ,σ] to [ξ,σ] is continuous. Moreover,

(10.2) [
∧
i∈I

fiτi,σ] =
∨
i∈I

(f∗i )
−[τi,σ]

for every family of surjective maps {fi : i ∈ I} with a common range.
If g : W → Z, then the map g∗ : WX → ZX is defined by g∗(h) = g ◦ h.

Let us observe that if g : σ → θ is a continuous map, then for every ξ the map
g∗ : [ξ,σ] → [ξ,θ] is also continuous. By (3.2),

(10.3) [ξ,σ] = [ξ,
∨

g∈C(σ,$)

g−$] =
∨

g∈C(σ,$)

g−∗ [ξ, $],

so that every continuous convergence with respect to a topology σ is the initial
convergence with respect to hyperconvergences. This fact implies that



402 Szymon Dolecki and Frédéric Mynard

Theorem 10.1. If for a given convergence ξ, the hyperconvergence [ξ, $] be-
longs to a concrete reflective subcategory J of convergences, then [ξ,σ] also
belongs to J for every topology σ.

A concrete reflective subcategory L of convergences is called Cartesian-closed
if [ξ,σ] ∈ L whenever ξ,σ ∈ L. The categories of convergences and of pseudo-
topologies are Cartesian-closed, but that of topologies is not.

Theorem 10.2. If L is a concrete reflective subcategory of convergences and
L is the corresponding reflector, then L is Cartesian-closed if and only if L
commutes with finite products.

Proof. Assume that L commutes with finite products and that ξ,σ ∈ L. By
definition, [ξ,σ] is the least convergence (on C(ξ,σ)) for which ξ× [ξ,σ] ≥ e−σ
(where e−σ stands for the initial convergence of σ by the natural evaluation).
Because σ ∈ L and thus e−σ ∈ L,

ξ ×L[ξ,σ] ≥ L(ξ × [ξ,σ]) ≥ L(e−σ) = e−σ,
hence L[ξ,σ] ≥ [ξ,σ] proving that [ξ,σ] ∈ L.

Conversely, suppose that L[ξ,σ] ≥ [ξ,σ] for every ξ,σ ∈ L. In order to prove
that

ξ ×Lτ ≥ L(ξ × τ)
for all convergences ξ and τ, it is enough to show that C(ξ × τ,σ) ⊂ C(ξ ×
Lτ,σ) for each σ = Lσ. If f ∈ C(ξ × τ,σ), then tf ∈ C(τ, [ξ,σ]) hence
tf ∈ C(Lτ,L[ξ,σ]) because L is a functor. It follows that

(tf)∗ ∈ C([L[ξ,σ],σ], [Lτ,σ]) ⊂ C([[ξ,σ],σ], [Lτ,σ]),
because [L[ξ,σ],σ] ≥ [[ξ,σ],σ] by virtue of the assumption. As the injection
i|ξ| is continuous from ξ to [[ξ,σ],σ], and (tf)∗ ◦ i|ξ| = f, we conclude that
f ∈ C(ξ ×Lτ,σ). �

Let X ⊂ Y,θ be a convergence on X and τ be a convergence on Y . If the
injection iX : X → Y is continuous, then we write θ B τ. It follows that
i∗X : C(τ,σ) → C(θ,σ) is continuous from [τ,σ] to [θ,σ]; on the other hand,
i∗X(f) = f ◦ iX, that is, i

∗
X is the restriction to X of maps on Y .

In the special case of σ = $, the restriction i∗X(A) is equivalent to the
intersection of a τ-closed set A ∈ C(τ, $) with X. Indeed, by definition,
i∗X(ψA)(x) = ψA(iX(x)), where ψA is the indicator function of a subset A
of Y , hence i∗X(ψA)(x) = 0 if and only if an element x of X belongs to A.
Accordingly, for a family A of θ-closed sets,

i∗∗X (A) = {A ∈ C(τ, $) : A ∈A}.
If ζ ≥ ξ, then C(ξ,σ) ⊂ C(ζ,σ) for every convergence σ, and the injection

is continuous from [ξ,σ] to [ζ,σ]. Let

(10.4) Epiσξ = i−|ξ|[[ξ,σ],σ],

where i|ξ| is the injection of |ξ| in C(C(ξ,σ),σ) (it is straightforward that i|ξ|
is continuous from ξ to [[ξ,σ],σ]).



Hyperconvergences 403

Consider now [·,σ] that associates with every convergence ξ on X, the power
convergence [ξ,σ]. On the other hand, [θ,σ] is a convergence on C(θ,σ) for
every convergence θ on a subset of |σ|X and each x ∈ X can be identified with
a map continuous from [θ,σ] to σ. Therefore, the couple of maps [·,σ], i−X[·,σ]
(where i−X[θ,σ] is the restriction of [θ,σ] to X) is a (−,−)-connection with
respect to the usual order on convergences at one end and to C on the other.

It follows from the general properties of Galois connections that [Epiσξ,σ] =
[ξ,σ] and that Epiσ is a projector. It is clear from the definition that Epiσ is
a concrete functor, hence it is a reflector.

11. Epitopologies.

By taking the supremum on the right-hand side of (10.4) over all topologies
σ, we define

Epiξ =
∨
σ=Tσ

Epiσξ =
∨
σ=Tσ

i−|ξ|[[ξ,σ],σ],

which in view of the preliminaries above is a concrete reflector. It is called
the epitopologizer and each convergence τ such that Epiτ = τ is called an
epitopology (or Antoine convergence).

It can be deduced from (10.3) that the epitopologizer can be represented by

Epiξ = i−|ξ|[[ξ, $], $],

thus Epiξ = Epi$ξ. Consequently, the epitopologizer is the composition of two
branches of the (−,−)-connection [·, $], i−X[·, $] (which is a special case of the
connection considered in the previous section). It follows, in particular, that

(11.1) [Epiξ, $] = [ξ, $]

and that Epiξ is the coarsest convergence ζ for which [ζ, $] = [ξ, $].

Proposition 11.1. The epitopologizer commutes with finite products:

(11.2) ξ × Epiτ ≥ Epi(ξ × τ).

Notice that (11.2) applied twice yields Epiξ × Epiτ ≥ Epi(ξ × τ).

Proof. As Epi is idempotent, it is enough to prove that ξ×Epiτ ≥ Epi(ξ×τ).
By virtue of the exponential law (10.1) and (11.1),

[ξ × Epiτ, $] = [ξ, [Epiτ, $]] = [ξ, [τ, $]] = [ξ × τ, $].
Therefore, ξ × Epiτ ≥ Epi(ξ × Epiτ) = Epi(ξ × τ). �

Hence, in view of Theorem 10.2, the category of epitopologies is Cartesian-
closed. On the other hand, every topology is an epitopology. Indeed, denote
by ιY the discrete topology on Y . Because ιC(ξ,$) ≥ [ξ, $], the injection from
C([ξ, $], $) to C(ιC(ξ,$), $) is continuous from [[ξ, $], $] to [ιC(ξ,$), $], and since
the topologizer T can be characterized via

Tξ =
∨

f∈C(ξ,$)

f−$ = i−|ξ|[ιC(ξ,$), $],



404 Szymon Dolecki and Frédéric Mynard

it is clear that Epiξ ≥ Tξ for every convergence ξ, which proves the claim.
We say that a convergence is a prototopology if limξF∩ limξG ⊂ limξ(F∧G)

for every couple of filters F,G. It is immediate that prototopologies constitute
a Cartesian-closed reflective subcategory of convergences.

Theorem 11.2. [4] The category of epitopologies is the least Cartesian-closed
reflective subcategory of prototopologies that includes all topologies.

Proof. Let L be a Cartesian-closed reflective subcategory of prototopologies
that contains all topologies. For every prototopology ξ, there exist a family
{τi : i ∈ I} of topologies and maps fi : |τi| → |ξ| such that ξ =

∧
i∈I fiτi. By

virtue of (10.2),

[ξ, $] =
∨
i∈I

(f∗i )
−[τi, $].

Because L is Cartesian-closed and contains all topologies, [τi, $] ∈ L for every
i ∈ I, and since L is reflective [ξ, $] ∈ L as the initial object with respect to
prototopologies in L. Therefore [[ξ, $], $] ∈ L by the Cartesian-closedness of L,
and thus the initial prototopology Epiξ = i−|ξ|[[ξ, $], $] belongs to L, because L
is reflective. It follows that every epitopology belongs to L. �

We shall now characterize epitopologies internally. We say that ξ is point-
diagonal if

(11.3) limξF ⊂ limξNξ•(F)

for every filter F. If ξ is a pseudotopology, then (11.3) is equivalent to

(11.4) adhξcl
\
ξ•H⊂ adhξH

for every filter H. Indeed, if x ∈ adhξcl
\
ξ•H and ξ is point-diagonal, then there

is a filter F#cl\ξ•H (equivalently, H#Nξ•(F)) and such that x ∈ limξF ⊂
limξNξ•(F), and thus x ∈ adhξH. Conversely, if (11.4) holds, ξ is a pseudo-
topology, and x /∈ limξNξ•(F), then there is a filter H#Nξ•(F) (equivalently,
F#cl\ξ•H) and such that x /∈ adhξH, hence by (11.4) x /∈ adhξcl

\
ξ•H thus

x /∈ limξF.
It follows from (12.2) (to be proved later) that

(11.5) limEpiξF =
⋂

cl
\

ξ•S3H#F

clξ(adhξH).

Therefore,

Theorem 11.3. A convergence ξ is an epitopology if and only if it is a point-
diagonal pseudotopology with closed limits.

Proof. Let ξ be an epitopology. Then it is a pseudotopology, because limSξF =⋂
H#F adhξH ⊂ limEpiξF. The closedness of limits follows from (11.5 ). If

x /∈ limξNξ•(F), then by (11.5) there exists a ξ-reduced filter H#Nξ•(F) such



Hyperconvergences 405

that x /∈ clξadhξH . Because H#Nξ•(F) is equivalent to cl
\
ξ•H#F and since

H = cl\ξ•H, also H#F, hence x /∈ limξF.
Conversely, let ξ be a point-diagonal pseudotopology with closed limits, and

let x /∈ limξF = limξNξ•(F). As ξ is a pseudotopology, there is an ultrafilter
U#Nξ•(F) (equivalently, F#cl

\
ξ•U) and such that x /∈ adhξU = adhξcl

\
ξ•U.

Because U is an ultrafilter and ξ has closed limits, adhξU = limξU = clξadhξH
where H = cl\ξ•U is a reduced filter which meshes with F. In view of (11.5),
x /∈ limEpiξF. �

It follows from (11.5) and (11.4) that

(11.6) clξ(adhξ(cl
\
ξ•H)) = clξ(adhEpiξH).

This formula which will be instrumental in several arguments, is a special case
of (12.3) that we will prove later.

Since ξ• is a principally based (called also finitely generated ) topology, and
Nξ•(x) = clξ∗x, the filter Nξ•(F) is generated by {clξ∗F : F ∈ F}, that is,
Nξ•(F) = cl

\
ξ∗F. Therefore the condition (11.3) is equivalent to

limξF ⊂ limξcl
\
ξ∗F,

and thus

Corollary 11.4. [4] A convergence ξ is an epitopology if and only if it is a
star-regular pseudotopology with closed limits.

Proposition 11.5. Every hyperconvergence is an epitopology.

Proof. From i−|ξ|[[ξ, $], $] ≥ [[ξ, $], $] it follows that [i
−
|ξ|[[ξ, $], $], $] C [[[ξ, $], $], $],

hence [ξ, $] = [i−|ξ|[[ξ, $], $], $] ≤ i
−
C(ξ,$)

[[[ξ, $], $], $] = Epi[ξ, $]. �

12. Duality.

Our goal is to characterize various properties of hyperconvergences in terms
of primal convergences, or rather in terms of equivalence classes of primal
convergences which correspond to the same hyperconvergence. As for every
convergence ξ the epitopologization Epiξ of ξ is the coarsest convergence such
that [Epiξ, $] = [ξ, $], primal characterizations of sundry properties of hyper-
convergences are naturally formulated as properties of epitopologies.

If M is a concrete functor such that M ≥ T, then C(Mτ,σ) ⊂ C(τ,σ)
for every convergence τ and each topology σ. Thus i|ξ| is an injection into
C(M[ξ, $], $) for every convergence ξ. Therefore one can define [17][14] the
M-epitopologizer by

EpiMξ = i
−
|ξ|([M[ξ, $], $]).

We say that ξ is an M-epitopology if ξ = EpiMξ. Because $ is an initially dense
object of the category of topologies,

(12.1) EpiMξ ≥ i
−([M[ξ,σ],σ])

for every topology σ (here i stands for the natural injection of |ξ| to C(C(ξ,σ),σ)).



406 Szymon Dolecki and Frédéric Mynard

As EpiM is a composition of two branches of a (−,−)-connection and of M
(inserted between them), we refer to this situation as to modified duality. It
follows that if M is a coreflector, then EpiM is a reflector; however, if M is a
reflector, then EpiM need not be a coreflector (in the category of convergences).

We can describe explicitly the convergence of filters for EpiMξ (a filter F
converges to x in EpiMξ if and only if for every ξ-closed set A with x /∈ A and
each ξ-hyperfilter G such that rG meshes with F implies that A /∈ limM[ξ,$]G).
However significant interpretations of this general description will be given
for special cases, where J and E are $-compatible classes of filters, and M is
either the reflector AJ, or the coreflector BE, or the composition AJBE (which
is in general neither reflector nor coreflector). In this section we shall merely
consider the reflector EpiBE which will be instrumental in the study of EpiAJBE
that we will undertake in subsequent sections.

A class E of filters is said to be $-compatible whenever G ∈ E implies that
rG ∈ E and if H ∈ E implies that e\ξH ∈ E. The classes of all filters, of
countably based filters, of countably deep filters (a filter F is countably deep
if
⋂
A∈A

A ∈ F for every countable A⊂F) and of principal filters are all $-

compatible.
According to our notation cl\ξ•E stands for the class of ξ-reduced E-filters.

In particular, cl\ξ•S is the class of ξ-reduced filters, cl
\
ξ•Pω of countably based ξ-

reduced filters, and cl\ξ•P of principal ξ-reduced filters. We notice that BE[ξ, $]
is based in ξ-saturated filters for every convergence ξ, but AE[ξ, $] is not in
general.

Proposition 12.1. [16] If E is $-compatible, then a filter F converges to x in
EpiBEξ if and only if x ∈ clξ(adhξH) for every filter H ∈ cl

\
ξ•E that meshes

with F.

In other words,

(12.2) limEpiB
E
ξF =

⋂
cl
\

ξ•E3H#F

clξ(adhξH),

Proof. Because BE[ξ, $] is based in ξ-saturated E-filters, a filter F converges to
x in EpiBEξ if and only if adhξrG \A 6= ∅ for every ξ-closed set A with x /∈ A
and each ξ-saturated E-filter G such that rG meshes with F, equivalently, if
x ∈ clξ(adhξrG) for every ξ-saturated E-filter G such that rG meshes with
F. Because of the duality between ξ-saturated and ξ-reduced E-filters for a $-
compatible class E, this means that x ∈ clξ(adhξH) for every ξ-reduced E-filter
H that meshes with F. �

Since cl\ξ•E ⊂ E and, of course, adhξH ⊂ clξ(adhξH), we infer that AEξ ≥
EpiBEξ for every convergence ξ. Therefore every BE-epitopology is adherence-
determined by E-filters. In particular we recover the already established fact
that each epitopology is a pseudotopology. Another already established fact



Hyperconvergences 407

that each BE-epitopology is an epitopology, is also an immediate consequence
of (12.2).

Let us observe that EpiBPξ = Tξ, that is, EpiBP is the topologizer. Indeed,
in this case (12.2) becomes

⋂
clξ•H∈F#

clξ(adhξH) which is equal to
⋂

H∈F#
clξH.

Therefore, every topology is a BE-epitopology under the provision that E in-
cludes P, the class of principal filters.

We notice that

(12.3) clξ(adhξ(cl
\
ξ•H)) = clξ(adhEpiB

E
ξH),

for every E-filter H. Indeed, if x ∈ adhEpiB
E
ξH then there exists a filter F that

meshes with H and converges to x for EpiBEξ, hence cl
\
ξ•H meshes with F, thus

by (12.2), x ∈ clξ(adhξ(cl
\
ξ•H)). On the other hand, by (11.4) adhξ(cl

\
ξ•H) ⊂

adhEpiB
E
ξH because EpiBEξ is an epitopology. Therefore, if B = Oξ(B), then

(12.4) adhEpiB
E
ξH#B ⇐⇒ adhξ(cl

\
ξ•H)#B

for every E-filter H.

13. Compactoidness.

It turns out that some properties of hyperconvergences can be rephrased in
terms of some compactness-like properties of their underlying convergences.

The concept of a set relatively compact with respect to another set has been
extended by various authors (see [8] for a historical account) to a filter (and,
more generally, family of sets) relatively compact with respect to a set, and in
[9], with respect to another family.

Let F be a class of filters. If A,B are families of subsets of |ξ|, then A is
said to be F-compactoid in B (for ξ) if adhξF ∈B# for every F ∈ F such that
F#A. A family A is F-compact if it is F-compactoid in itself. In particular,
if A ⊂ X, then {A} (equivalently, A = {A}↑) is compactoid in X (for a
convergence on X) if and only if A is relatively compact; {A} (equivalently,
A = {A}↑) is compact if and only if A is compact. Compactoidness is a common
generalization of compactness (of sets) and of convergence (of filters). Indeed,
a filter F is compactoid in {x} for ξ if and only if x ∈ limSξF, where S is the
pseudotopologizer.

This notion of compactoidness needs to be ultimately extended. Let F and
G be classes of filters, and let ξ and τ be convergences on a common set. A
family A is said to be F

G
-compactoid in B (for ξ

τ
) [8, Section 8] if

(13.1) ∀F∈F∀G∈G F ≥G, adhτG ∈A# =⇒ adhξF ∈B#.
If G = P is the class of principal filters and τ = ι is the discrete topology,
then ∀G∈P F ≥G, adhιG ∈A# is equivalent to F#A, and the whole condition
(13.1) becomes

∀F∈FF#A =⇒ adhξF ∈B#,
that is, to the F-compactoidness of A in B for ξ.



408 Szymon Dolecki and Frédéric Mynard

The use of two classes of filters enables one to recover such compactness-like
notions like Lindelöf property, which is in these terms S

Pω
-compactoidness.

Another important special case of (13.1) is that of cover compactoidness.
A family P of subsets of |ξ| is a ξ-cover of a set A if every filter such that
limξF ∩A 6= ∅ contains an element of P; a family P is an ideal if Q ⊂ P ∈P
implies that Q ∈ P and if

⋃
R ∈ P for every finite subfamily R of P. It was

observed in [8] that a family P is a ξ-cover of A if and only if adhξPc ∩A = ∅
(Recall that Pc = {Pc : P ∈P}). It follows that

Theorem 13.1. [8] A family A is F
G

-compactoid in B (for ξ
τ

) if and only if
for every B ∈ B and each ξ-cover P of B such that Pc ∈ F there exist A ∈ A
and a refinement R of P which is a τ-cover of A such that Rc ∈ G.

If ξ = τ in the theorem above, then we say that A is F
G

-cover-compactoid in
B for ξ. For example, A is S

Pω
-cover-compactoid for ξ in B if for every (ideal)

ξ-cover P of B ∈B there exist A ∈A and a countable subfamily R of P which
is a ξ-cover of A. If ξ is a topology, then ideal can be dropped without altering
the meaning [8], hence this means that A is Lindelöf in B for ξ.

Let us recall that cover-compactoidness and compactoidness coincide for
topologies, but do not for arbitrary convergences.

At first we notice that convergence with respect to the epitopologization is
a compactoidness property. More generally,

Proposition 13.2. If E is $-compatible, then a filter converges to x in EpiBEξ
if and only if it is cl\ξ•E-compactoid in Nξ(x) for ξ.

Proof. On rephrasing Proposition 12.1, a filter F converges to x in EpiBEξ if
and only if adhξH∩ O 6= ∅ for every ξ-open set O that contains x and each
ξ-reduced filter H∈ E that meshes with F. �

In particular,

Proposition 13.3. A filter converges to x in Epiξ if and only if it is cl\ξ•S-
compactoid in Nξ(x) for ξ.

In the sequel we shall make repeated use of the following immediate conse-
quence of (12.4):

Lemma 13.4. [11, Lemma 6.3] Let E be a $-compatible class of filters. If B is
a family of ξ-open sets, then a family is E-compactoid in B for EpiBEξ if and
only if it is cl\ξ•(E)-compactoid in B for ξ.

It follows from Proposition 13.2 and from Lemma 13.4 that

Corollary 13.5. A filter converges to x in EpiBEξ if and only if it is E-
compactoid in Nξ(x) for Epiξ.



Hyperconvergences 409

14. Topological core compactness and commutativity with finite
products.

Given a subcategory L of convergences, a convergence ξ is called L-exponential
if [ξ,σ] ∈ L for every σ ∈ L. It follows immediately from the definitions that
a subcategory is Cartesian-closed if and only each of its objects is exponen-
tial. By Theorem 10.2, a concrete reflective subcategory of convergences is
Cartesian-closed if and only if the corresponding reflector commutes with finite
products. In fact more is true, namely, a convergence ξ is L-exponential if and
only if

ξ ×Lτ ≥ L(ξ × τ)
for every convergence τ (where L is the reflector corresponding to L).

It is known [18] that a topology is T-exponential if and only if it is core com-
pact, that is, for every x and each neighborhood O of x there is a neighborhood
V of x which is compactoid in O [13]. We shall refine this classical result.

Let J and E be classes of filters. The following functor QJ
E

in the category
of convergences was defined in [17]:

x ∈ lim
Q
J

E
ξ
F

whenever for every O ∈ Oξ(x) there exists a filter J ∈ J such that J ≤ F
and J is E-compactoid in O for ξ. We say that a convergence ξ is topologically
J-core E-compact at x if x ∈ limξF implies that x ∈ limSQJ

E
ξ
F. If J = P, the

class of principal filters, then we abridge P-core to core. Let us observe that

(14.1) SQP
E

= QP
E
.

Indeed, if x ∈ limSQP
E

F then for every ultrafilter U finer than F and each
O ∈ Oξ(x) there exists FU ∈ U which is E-compactoid in O for ξ. Therefore
there exists n < ω and U1,U2, . . . ,Un such that F = FU1 ∪FU2 ∪ . . .∪FUn ∈F
and obviously F is E-compactoid in O for ξ.

A class G of filters is composable if HA ∈ G(Y ) for every H ∈ G(X × Y )
and A∈ G(X) and if it includes the class of principal filters. Because the star
and the point closures can be defined as relations, we conclude that

Proposition 14.1. Every composable class is $-compatible.

Theorem 14.2. [17, Theorem 6.2] If E and J are composable classes of filters,
then

(14.2) θ × AJτ ≥ T(ξ × τ)
for every τ = BEτ if and only if

(14.3) θ ≥ SQJ
E
EpiBEξ.

Proof. Assume (14.3) and τ = BEτ. Since the reflector S commutes with
the product, we can assume without loss of generality that θ ≥ QJ

E
EpiBEξ.

Consider filters F and G such that x ∈ limθF and y ∈ limAJτG and a (ξ × τ)-
closed set H that meshes with F × G. We will show that (x,y) ∈ H and
thus (14.2) holds. By (14.3) for every W ∈ Oξ(x) there exists a J-filter



410 Szymon Dolecki and Frédéric Mynard

DW ≤ F which is E-compactoid in W for EpiBEξ. It follows that HDW #G
and HDW ∈ J because J is composable. Hence y ∈ adhτHDW and thus
there exists an E-filter LW #HDW such that y ∈ limτLW . Therefore DW
meshes with the E-filter H−LW (because E is composable). We infer that
there is xW ∈ adhEpiB

E
ξH
−LW ∩W as DW is E-compactoid in W for EpiBEξ,

and thus (xW ,y) ∈ H, because T(EpiBEξ × τ) = T(ξ × τ). Accordingly,
x ∈ clξH−y = H−y, that is, (x,y) ∈ H.

Conversely, if (14.3) does not hold, then there exist x0 and a free ultrafilter
U such that x0 ∈ limθU and there is W ∈ Oξ(x0) such that for every J-
filter J ≤ U, there exists an E-filter EJ that meshes with J and such that
adhEpiB

E
ξEJ∩W = ∅. Denote by J(U) the set of J-filters which are coarser than

U. Let τ be the convergence on X for which every point is isolated except for x0,
and a free filter G converges to x0 if there is J in J(U) for which G ≥EJ . This is
an E-based convergence and x0 ∈ limAJτU thus (x0,x0) ∈ limθ×AJτ (U×U). The
filter U×U meshes with the set A = {(x,y) : x ∈ limEpiB

E
ξ{y}↑,y 6= x0}. Let C

be a filter on A such that (x,y) ∈ limξ×τC. If y 6= x0 then (x,y) ∈ A, because
y is isolated in τ and limEpiB

E
ξ{y}↑ is ξ-closed for every y. If y = x0 then there

exist filters J ∈ J(U) and F such that x ∈ limξF and F ×EJ ≤C. Obviously
A meshes with F ×EJ , that is, for each F ∈F and every E ∈ EJ there exist
x ∈ F and y ∈ E with x ∈ limEpiB

E
ξ{y}↑ = cl(EpiB

E
ξ)∗y. Therefore cl

\
(EpiB

E
ξ)∗
F

meshes with EJ . Because EpiBEξ is star-regular, x ∈ limEpiB
E
ξcl

\
(EpiB

E
ξ)∗
F,

hence x ∈ adhEpiB
E
ξEJ and hence (x,x0) belongs to the (ξ × τ)-closed set

Wc×{x0}. It follows that (x0,x0) /∈ A∪Wc×{x0}⊃ clξ×τA so that (x0,x0) /∈
limT(ξ×τ)(U ×U). �

It is obvious that

Proposition 14.3. (14.2) holds for every τ = BEτ if and only if

θ × AJBEτ ≥ T(ξ × τ)

for every convergence τ.

15. Reflective properties of hyperconvergence.

It has already been mentioned that a hyperconvergence associated with a
topology need not be a pretopology (a fortiori need not be a topology). Topo-
logicity (Tτ ≥ τ) and pretopologicity (Pτ ≥ τ) are examples of reflective
properties. We shall provide a general characterization of a reflective property
of hyperconvergences in terms of a coreflective property of primal convergences.
If J is a property of convergences, then by definition ξ ∈ J∗ whenever [ξ, $] ∈ J.
If J is a concrete reflective subcategory of convergences, then by Theorem 10.1,
ξ ∈ J∗ if and only if [ξ,σ] ∈ J for every topology σ.

Proposition 15.1. If J is a concrete reflective subcategory, then J∗ is a (con-
crete) coreflective subcategory.



Hyperconvergences 411

Proof. Let fi : |ξi|→ Y and ξi ∈ J∗ for every i ∈ I, that is, [ξi, $] ∈ J for each
i ∈ I. Then by (10.2), [

∧
i∈I fiξi, $] =

∨
i∈I(f

∗
i )
−[ξi, $], and since by assump-

tion J is a concrete reflective subcategory of convergences,
∨
i∈I(f

∗
i )
−[ξi, $] ∈ J,

thus
∧
i∈I fiξi ∈ J∗. �

Theorem 15.2. [17] If M ≥ T is a concrete functor, then the following state-
ments are equivalent:

M[ξ, $] ≥ [ξ, $];(15.1)
ξ ≥ EpiMξ;(15.2)

ξ ×Mτ ≥ T(ξ × τ) for every τ.(15.3)

Proof. If M[ξ, $] ≥ [ξ, $], then [[ξ, $], $] ≥ [M[ξ, $], $] so that ξ ≥ Epiξ ≥
EpiMξ. Conversely if ξ ≥ EpiMξ then by (12.1) the injection i : ξ → [M[ξ,σ],σ]
is continuous. By (10.1), the evaluation

ti : ξ ×M[ξ,σ] → σ

is also continuous. But [ξ, $] is the coarsest convergence on C(ξ, $) that makes
the evaluation continuous. Therefore, M[ξ, $] ≥ [ξ, $].

Assume (15.1) and let f ∈ C(ξ × τ,σ). Then by (10.1) and because M is a
functor, tf ∈ C(τ, [ξ,σ]) = C(Mτ,M[ξ,σ]) ⊂ C(Mτ, [ξ,σ]) by (15.1). Again
by (10.1 ), f ∈ C(Mτ, [ξ,σ]) = C(ξ ×Mτ,σ). Therefore (15.3) holds.

The hyperconvergence [ξ, $] is the coarsest convergence τ for which the nat-
ural evaluation is continuous from ξ×τ to $. Therefore if (15.3) holds, then the
evaluation is also continuous from ξ×M[ξ, $] to $ and thus M[ξ, $] ≥ [ξ, $]. �

It follows from Theorems 15.2 and 14.2 and from Proposition 14.3 that

ξ ≥ EpiAJBEξ ⇐⇒ ξ ≥ SQ
J

E
EpiBEξ

(but this does not mean that EpiAJBE = SQ
J

E
EpiBE). This property of ξ is

equivalent to a (reflective) property of [ξ, $] = [Epiξ, $], hence it is a property
of epitopologies. We conclude that

Corollary 15.3. If E and J are composable classes of filters, then AJBE[ξ, $] ≥
[ξ, $] if and only if Epiξ is topologically J-core E-compact for EpiBEξ (equiva-
lently ξ is J-core cl\ξ•E-compact).

Consider now several special cases of the condition above. As BS is the
identity and AS is the pseudotopologizer, and since every hyperconvergence
is a pseudotopology, the condition AS[ξ, $] ≥ [ξ, $] is always fulfilled. On the
other hand, by Corollary 15.3, if x ∈ limξF then adhEpiξ H∩W 6= ∅ for every
W ∈ Oξ(x) and for each ultrafilter U finer than F and for every filter H that
meshes with U; in other words, x ∈ clξ(adhEpiξH), which is of course always
the case.

We apply now Corollary 15.3 in the case E = S (the class of all filters) and
J = Pω (the class of countably based filters).



412 Szymon Dolecki and Frédéric Mynard

Theorem 15.4. The hyperconvergence [ξ, $] is a paratopology if and only if
Epiξ is topologically Pω-core compact.

If E = S is the class of all filters and J = P is the class of principal filters,
then Corollary 15.3 yields

Theorem 15.5. The hyperconvergence [ξ, $] is a pretopology if and only if Epiξ
is topologically core compact.

Thanks to Theorem 10.1 we infer that

Corollary 15.6. If an epitopology ξ is topologically core (respectively, Pω-
core) compact, then [ξ,σ] is pretopological (respectively, paratopological) for
each topology σ.

Corollary 15.3 cannot be used to characterize the topologicity of hypercon-
vergences, because T = AT, but the class T of principal filters of closed sets
is not composable; instead one can apply directly Theorem 15.2 to another
representation of the topologizer [11]. However, in the next section we will use
yet another approach.

16. Vicinities and neighborhoods.

Certain objects related to hyperconvergences, like vicinity or neighborhood,
can be characterized in terms of primal convergences. This enables one to
characterize, in terms of primal convergences, various reflective and coreflective
properties of hyperconvergences that can be formulated with the aid of such
objects.

Some of the reflective properties of hyperconvergences have been charac-
terized already by virtue of Theorem 15.2. The situation is different for the
coreflective properties of hyperconvergences, because for them there exists no
equivalence scheme similar to Theorem 15.2.

In order to characterize hypervicinities and open hypersets, we introduce a
notion of rigid compactoidness. A family H ξ-rigidly meshes with a family A
if for every H ∈ H, there is B = clξB ⊂ H such that B ∈ A#. Of course, if
H ξ-rigidly meshes with A, then it meshes with A. On the other hand, if H is
ξ-closed (that is, H = cl\ξH) and meshes with A, then it ξ-rigidly meshes with
A.

A family A is rigidly E-compactoid in B for ξ if adhξE ∈B# for each E-filter
E which ξ-rigidly meshes with A; a family is rigidly E-compact if it is rigidly
E-compactoid in itself. Every E-compactoid (E-compact) family is rigidly E-
compactoid (rigidly E-compact). If ξ is a topology, then the converse also
holds, because then for every filter H, one has adhξH =

⋂
H∈H clξH and thus

adhξH = adhξcl
\
ξH.

Lemma 16.1. A set F is cl\ξ•S-compactoid in B for ξ if and only if Oξ(F) is
rigidly compactoid in B for ξ.



Hyperconvergences 413

Proof. We notice that H rigidly meshes with Oξ(F) if and only if int
\
ξ∗H meshes

with F. Indeed, the former means that for each H ∈ H there is a subset
D = clξD of H such that D ∈ Oξ(F)#, equivalently clξD ∩ F 6= ∅, or else
int\ξ∗H meshes with F.

Therefore if Oξ(F) is rigidly compactoid in B for ξ and int
\
ξ∗H = H = cl

\
ξ•H

meshes with F, then H rigidly meshes with Oξ(F) and thus adhξH∩B 6= ∅.
Conversely if F is cl\ξ•S-compactoid in B for ξ and H rigidly meshes with Oξ(F),
then the ξ-reduced filter int\ξ∗H meshes with F and thus adhξ(int

\
ξ∗H) ∩B 6=

∅. �

It is straightforward that each intersection of stable hyperfilters is stable; in
particular, each vicinity filter for a hyperconvergence is stable. On the other
hand, an open hyperset (that is, open for a hyperconvergence) is always stable.
Observe that A is ξ-stable if and only if Ac = Oξ(Ac). Let us characterize
stable vicinities.

Theorem 16.2. Let E be $-compatible. A hyperset A is a stable vicinity of A0
for BE[ξ, $] if and only if Ac = Oξ(Ac) is rigidly E-compactoid in Ac0 for ξ.

Proof. A hyperset A is a vicinity of A0 for BE[ξ, $] if and only if A∈ E for every
E-hyperfilter E that [ξ, $]-converges to A0. Equivalently, if for each ξ-reduced
E-filter E with adhξE ⊂ A0 there exists E ∈E such that if B = clξB ⊂ E, then
B ∈ A. In other words, for every E ∈ E there exists B = clξB ⊂ E such that
B /∈ A#c , then adhξE∩Ac0 6= ∅; equivalently, Ac is rigidly cl

\
ξ•E-compactoid in

Ac0 for ξ. �

Corollary 16.3. Let E be $-compatible. A hyperset A is open for BE[ξ, $] if
and only if Ac = Oξ(Ac) and is rigidly E-compact for ξ.

This refines a result of [11, Corollary 5.3]; if ξ is a topology, in the case
where E is the class of all filters this recovers [10, Theorem 3.1] of Dolecki,
Greco and Lechicki, and for the class E of countably based filters [2, Theorem
2.1] of Alleche and Calbrix.

We will use the characterizations above to describe the pretopologicity and
the topologicity of hyperconvergences in terms of primal convergences. The
theorem below is identical with Theorem 15.5; what differs is the proof.

First observe that, as every hyperconvergence has a convergence base con-
sisting of saturated filters, every vicinity filter has a filter base consisting of
saturated, hence stable, hypersets.

Theorem 16.4. The hyperconvergence [ξ, $] is a pretopology if and only if Epiξ
is topologically core compact.

Proof. By definition, [ξ, $] is a pretopology if and only if A0 ∈ lim[ξ,$]V[ξ,$](A0),
where V[ξ,$](A0) denotes the filter of vicinities of A0 for [ξ, $]. In other words,
adhξrV[ξ,$](A0) ⊂ A0. Equivalently, if limξF ∩Ac0 6= ∅, then F does not mesh
with rV[ξ,$](A0), in other words, there is F ∈ F and a stable A ∈ V[ξ,$](A0)
such that F ∩

⋃
A∈AA = ∅, that is, F ⊂

⋂
A∈AA

c. By Theorem 16.2, this



414 Szymon Dolecki and Frédéric Mynard

means that for every ξ-open set O and each filter F such that limξF ∩O 6= ∅
there exists a family B = Oξ(B) which is rigidly compactoid in O for ξ and
such that

⋂
B∈B B ∈ F. It follows that ξ is topologically core cl

\
ξ•S-compact.

Indeed, as B is rigidly compactoid in O for ξ, and B ⊂Oξ(F), the latter is also
rigidly compactoid in O for ξ, thus by Lemma 16.1, F is cl\ξ•S-compactoid in O
for ξ. Conversely, if ξ is topologically core cl\ξ•S-compact and limξF ∩O 6= ∅,
then there exists F which is cl\ξ•S-compactoid in O for ξ, hence by Lemma
16.1 B = Oξ(F) is rigidly compactoid in O for ξ and of course

⋂
B∈B B ∈ F.

Finally, since O is ξ-open, by Lemma 13.4, it follows that F is compactoid in
O for Epiξ. �

A convergence is a topology if and only if the filter of neighborhoods of
every point converges to that point. Once again, as [ξ, $] = [Epiξ, $], we can
formulate any characterization of the pretopologicity of hyperconvergences in
terms of the underlying epitopologies.

Theorem 16.5. The hyperconvergence [ξ, $] is a topology if and only if for
every ξ-open set O and each filter F such that limξF ∩ O 6= ∅ there exists a
family B = Oξ(B) which is rigidly compact for Epiξ and such that O ∈ B and⋂
B∈B B ∈F.

As every compact family is rigidly compact, the existence of a compact
family B in the statement above is a sufficient condition for the topologicity of
[ξ, $] (and by Theorem 10.1, of [ξ,σ] for every topology σ).

In the special case where ξ is a topology, this is a condition of Day and
Kelly [6] (who did not use the name compact family, but gave an equivalent
definition) and the characterization of core compactness [13, Proposition 4.2]
by Hofmann and Lawson, if we remember that a family B = Oξ(B) is compact if
and only if it is an open set for the Scott topology associated with a topology ξ.
Therefore it must follow that if the primal convergence is a topology, then the
topologicity and the pretopologicity of the hyperconvergence coincide. This
is actually the case. We do not provide here the proof, because it uses the
technique of multifilters, and would require quite a few preliminaries.

Theorem 16.6. [11, Theorem 5.6] If ξ is a topology, then [ξ, $] is a topology
if and only if [ξ, $] is a pretopology.

Example 16.7. Consider the bisequence {x∞}∪{xn : n < ω}∪{xn,k : n,k <
ω} endowed with the canonical pretopology π: the filter associated with (xn)n
is the coarsest free filter that converges to x∞ and for every n < ω, the filter
associated with (xn,k)k is the coarsest free filter that converges to xn. This is a
Hausdorff locally compact pretopology (hence a point-diagonal pseudotopology
with closed limits), so that its hyperconvergence is pretopological in view of
Theorem 16.4. On the other hand, the topologization of π is Hausdorff regular,
but not locally compact, thus [Tπ, $] is not a pretopology.



Hyperconvergences 415

17. Coreflective properties of hyperconvergence.

Let us start by considering the case of coreflectors of the form M = BE,
where BE is defined in (9.4).

Theorem 17.1. Let E be a $-compatible class of filters. The hyperconver-
gence [ξ, $] is E-based if and only if every ξ-open set is S

E
-compactoid for ξ

(equivalently, for Epiξ).

Proof. Let [ξ, $] be E-based and let A be an ξ-closed set. To show that Ac is
S

E
-compactoid let G to be a filter such that adhξG ⊂ A, that is, A ∈ lim[ξ,$]e

\
ξ•G.

By the assumption there exists an E-filter E coarser than e\ξ•G for which A ∈
lim[ξ,$]E, hence adhξrE ⊂ A. The filter E = rE is a ξ-reduced E-filter, because
E is $-compatible, and E ≤ re\ξG. For every E ∈ E let GE ∈ G be such that
intξ∗GE ⊂ E. Then the filter H generated by {GE : E ∈ E} is an E-filter
coarser than G and than E and hence adhξH⊂ A.

Conversely if every ξ-open set is S
E

-compactoid and A ∈ lim[ξ,$]G, then
adhξrG ⊂ A, and thus there exists an E-filter E coarser than rG such that
adhξE ⊂ A, because Ac is S

E
-compactoid. Hence A ∈ lim[ξ,$]e

\
ξE and e

\
ξE

belongs to E, because E is $-compatible, and e\ξE ≤ e
\
ξrG ≤ G. �

As a corollary we get (for general convergences, open hereditarily cover Lin-
delöf need not entail hereditarily cover Lindelöf ).

Theorem 17.2. [15] The hyperconvergence [ξ, $] is of countable character if
and only if ξ (equivalently, Epiξ) is open hereditarily cover Lindelöf.

Observe that a convergence τ is respectively bisequential, strongly Fréchet,
Fréchet and sequential if and only if Sτ ≥ SFirstτ,Pωτ ≥ PωFirstτ, Pτ ≥
PFirstτ and Tτ ≥ TFirstτ.

In other words, a convergence is Fréchet whenever every sequential vicinity
of a point is a vicinity. Therefore,

Theorem 17.3. The hyperconvergence [ξ, $] is Fréchet if and only if every
family B = Oξ(B) which is rigidly countably compactoid in a ξ-open set O, is
rigidly ξ-compactoid in O.

A convergence is sequential whenever every sequentially open set is open.
Hence

Theorem 17.4. The hyperconvergence [ξ, $] is sequential if and only if every
family B = Oξ(B) which is rigidly countably ξ-compact is rigidly ξ-compactoid.

Recall that a topology τ is tight if x ∈ clτA implies the existence of a
countable subset E of A such that x ∈ clτE. As a special case of [15, Theorem
3.7], we have

Theorem 17.5. If ξ is a topology and if T [ξ, $] is tight, then ξ is hereditarily
Lindelöf.



416 Szymon Dolecki and Frédéric Mynard

Proof. Let G be a filter such that adhξG ⊂ A = clξA, that is, A ∈ lim[ξ,$]e
\
ξ•G.

In particular, A ∈ cl[ξ,$]{clξG : G ∈G}, and since T [ξ, $] is tight, there exists a
sequence (Gn) such that A ∈ cl[ξ,$]{clξGn : n < ω}, that is, for every ξ-compact
family B = Oξ(B) such that Ac ∈B there exists n < ω such that (clξGn)c ∈B.
It follows that adhξ(Gn) =

⋂
n<ω clξGn ⊂ A, for if x ∈

⋂
n<ω clξGn \A, then

(clξGn)c /∈ Oξ(x) for every n < ω and Ac ∈ Oξ(x). Of course, Oξ(x) is
Tξ-compact, which leads to a contradiction. �

Corollary 17.6. Let ξ be a topology. The following are equivalent:
(1) ξ is hereditarily Lindelöf;
(2) [ξ, $] is first-countable;
(3) [ξ, $] is countably-tight;
(4) [ξ, $] is sequential;
(5) T [ξ, $] is sequential;
(6) T [ξ, $] is countably-tight;

Recall that every perfectly normal Lindelöf topology is hereditarily Lindelöf
[12, Exercise 3.8.A]. Notice that, unlike in the cited book of Engelking, our
definition of Lindelöf convergence does not involve any separation axiom. On
the other hand, the atomic cofinite topology on an uncountable set is (Haus-
dorff) compact, hence Lindelöf, but not hereditarily Lindelöf (for the atomic
cofinite topology on X all elements but one are isolated, and a subset of X is
a neighborhood of the non isolated point whenever its complement is finite).
Indeed, the restriction of this topology to the (open) complement of the non
isolated element is the discrete topology on an uncountable set.

18. Lattice-theoretic approach to hyperconvergence.

If L is a complete lattice with respect to an order ≤, then three convergences
arise naturally: lower, upper, and their infimum. A filter F converges to x in
the upper convergence (x ∈ lim+F) if

lim supF =
∧
F∈F

∨
F ≤ x.

This formula obviously defines a convergence, because F ≤G implies lim supF ≥
lim supG, and x =

∨
F3x

∧
F. Clearly, lim supF ≤ x ≤ y implies that

lim supF ≤ y, that is, every limit is upper stable. It follows that every +-
closed set is upper stable, hence each +-open set is lower stable. By definition,
a subset O of L is +-open if

∧
F∈F

∨
F ≤ x for a filter F and x ∈ O then

O ∈ F; in other words, if
∧
A ∈ O then there is a finite subset B of A such

that
∧
B ∈ O.

Open sets for the lower convergence (that is, the upper convergence with
respect to the inverse order) were characterized by D. Scott [19] with the
aid of the relation of being way below: an element x (of a complete lattice)
is way below an element y (in symbols, x � y) whenever

∨
A ≤ y implies∨

F ≤ x for some finite subset F of A. Accordingly, a set O is open for the
lower convergence if x ∈ O and x ≤ y imply y ∈ O, and if

∨
A ∈ O, then there



Hyperconvergences 417

is a finite subset B of A such that
∨
B ∈ O. In the special case of the lattice

of open sets for a topology ξ, this means precisely that B is open for the lower
topology if and only if Oξ(B) = B is ξ-compact.

A complete lattice is called continuous if its lower convergence is topological,
that is, if each neighborhood filter converges to its defining point; in other
words, if y =

∨
{x : x � y} for every y ∈ L. In [13] Hofmann and Lawson

proved that the lattice of open sets of a topology ξ is continuous if and only
if ξ is core compact. This is dual to the result from preceding paragraphs
that a topology ξ is core compact if and only if the hyperconvergence [ξ, $]
is topological. Moreover, Hofmann and Lawson showed that in the class of
so-called sober topologies, core compactness is equivalent to hereditary local
compactness. We shall relate these facts in terms of hyperspaces.

The hyperspace C(ξ, $) ordered by inclusion is a complete lattice in which
the supremum and the infimum are given by∨

A = clξ(
⋃
A∈A

A);
∧
A =

⋂
A∈A

A.

If ξ is a topology, then adhξH =
⋂
H∈H clξH, hence A ∈ lim[ξ,$]F if and only

if
∧
F∈F

∨
F ⊂ A, that is, the hyperconvergence (of a topology) is the upper

convergence in the lattice of closed sets.
An element p of a lattice is prime if a∨ b ≥ p implies that either a ≥ p or

b ≥ p. A ξ-closed set is called irreducible if it is prime in the lattice C(ξ, $),
that is, if it cannot be written as the union of two proper ξ-closed subsets.
Notice that clξ{x} is irreducible for every x ∈ |ξ| and for each convergence ξ.
The whole space of the cofinite topology on an infinite set is an example of
irreducible closed set which is not the closure of a point. A T0 convergence
is called sober if every irreducible closed set contains a dense point. Each
Hausdorff topology is sober; the cofinite topology on an infinite set is a T1 non
sober topology. Being sober is of course a topological property.

The lower Vietoris (or lower Kuratowski ) topology V−(ξ) on C(ξ, $) has
the following subbase of open sets: O− = {A ∈ C(ξ, $) : A ∩ O 6= ∅} where
O ∈Oξ. Notice that for a hyperset A, a hyperpoint A0 ∈ clV−(ξ)A if and only
if A0 ⊂ clξ(

⋃
A∈AA), that is, whenever A0 ≤

∨
A in the lattice C(ξ, $).

The spectrum Σ(ξ) of C(ξ, $) is the set of non empty irreducible ξ-closed sets,
endowed with the restriction of the lower Vietoris topology. In other words, if
we let cl\ξU = {clξ{x} : x ∈ U}, then the topology of Σ(ξ) is generated by the
hypersets Σ(ξ) \ cl\ξA with A ∈ C(ξ, $).

In our terms, [13, Proposition 2.7] implies that

Proposition 18.1. The map clξ : |ξ| → Σ(ξ) is continuous and open on its
image; it is injective if and only if it is an embedding if and only if ξ is T0; it
is bijective if and only if it is a homeomorphism if and only if ξ is sober.

Proof. The first assertion follows from the fact that if O is ξ-open, then clξ{x}∈
O− whenever clξ{x}∩O 6= ∅, that is, whenever x ∈ O. To prove the second
statement, notice that ξ is T0 if and only if x0 6= x1 implies clξ{x0} 6= clξ{x1}.



418 Szymon Dolecki and Frédéric Mynard

The third statement is a consequence of the fact that ξ is sober if and only if
the only non empty irreducible ξ-closed sets are the closures of singletons. �

We prove here directly the following conclusion of [13, Proposition 2.7].

Proposition 18.2. The spectrum is sober.

Proof. Let A ⊂ Σ(ξ) be a V−(ξ)-closed hyperset (that is, {A ∈ Σ(ξ) : A ⊂
clξ(
⋃
B∈AB)}) which is not the closure of a hyperpoint of Σ(ξ); in other words,

the set clξ(
⋃
B∈AB) is reducible: there exist ξ-closed sets A0,A1 such that

clξ(
⋃
B∈AB) = A0 ∪ A1 and A1 \ A0 6= ∅ and A0 \ A1 6= ∅. The hypersets

A0 = {A ∈ Σ(ξ) : A ⊂ A0} and A1 = {A ∈ Σ(ξ) : A ⊂ A1} are V−(ξ)-closed.
Let A ⊂ A0 ∪ A1, where A is an element of Σ(ξ). As A is irreducible, either
A∩A0 is a subset of A∩A1 and thus A ⊂ A1 or vice versa and thus A ⊂ A0.
This shows that A is reducible and hence Σ(ξ) endowed with V−(ξ) is sober. �

It follows that every T0 topology ξ can be embedded in a sober topology
(the sobrification of ξ).

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Received February 2002

Revised September 2002

Szymon Dolecki

Département de Mathématiques, Université de Bourgogne, B. P. 47870, 21078
Dijon, France

E-mail address : dolecki@u-bourgogne.fr

Frédéric Mynard

Department of Mathematics, Hume Hall 308, University of Mississippi, Uni-
versity, MS 38677, USA

E-mail address : mynard@olemiss.edu