@ Appl. Gen. Topol. 15, no. 2(2014), 121-136doi:10.4995/agt.2014.3156
c© AGT, UPV, 2014

Convergence S-compactifications

Bernd Losert and Gary Richardson

Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

(berndlosert@knights.ucf.edu, gary.richardson@ucf.edu)

Abstract

Properties of continuous actions on convergence spaces are investigated.

The primary focus is the characterization as to when a continuous ac-

tion on a convergence space can be continuously extended to an action

on a compactification of the convergence space. The largest and small-

est such compactifications are studied.

2010 MSC: Primary 54A20; Secondary 54D35.

Keywords: convergence space; convergence semigroup; continuous action;
S-compactification.

1. Introduction and Preliminaries

The study of the notion of a topological transformation group or G-space
dates back to the work of Gottshalk and Hedlund [4] who generalized several
classical dynamical results from the theory of differential equations and other
fields of mathematics. The problem of characterizing when a G-space has a
compactification where the action can be continuously extended to the com-
pactification, called a G-compactification, seems to have originated with de
Vries [3]. The term “S-compactification” is used in the semigroup context.
Our work is devoted to the study of S-compactifications where the underlying
spaces are convergence spaces rather than topological spaces. It is known that
the category of convergence spaces has nicer categorical properties. For exam-
ple, quotient maps are productive in the category of convergence spaces but not
in the category of topological spaces. Some results on S-spaces in the context
of convergence spaces can be found in [1, 2]. A good reference on convergence
spaces and categorical terminology is the book by Preuss [11].

Received 10 February 2013 – Accepted 30 January 2014

http://dx.doi.org/10.4995/agt.2014.3156


B. Losert and G. Richardson

Let X be a set, P(X) the power set of X, and F(X) the set of all filters on
X. For x ∈ X, ẋ denotes the fixed ultrafilter on X generated by {{x}}. Define
the following partial order on F(X): Given F, G ∈ F(X), F ≥ G (read “F is
finer than G”) if and only if G ⊆ F. Given F, G ∈ F(X), the least upper bound
F ∨ G of F and G exists provided that F ∩ G 6= ∅ for each F ∈ F and G ∈ G
and it is the smallest filter containing both F and G.

Definition 1.1. A pair (X, q) is called a convergence space whenever X is
a set and q : F(X) → P(X) obeys:

(CS1) x ∈ q(ẋ)
(CS2) G ≥ F implies q(F) ⊆ q(G)
(CS3) x ∈ q(F) implies x ∈ q(F ∩ ẋ)

A function q obeying (CS1) through (CS3) is called a convergence struc-
ture on X. The notation x ∈ q(F) is read as “F q-converges to x” or as “F

converges to x” and is usually written as F
q
→ x or F → x. Most of the time

we will not need to make explicit reference to the convergence structure so we

will normally write (X, q) as X and F
q
→ x as F → x.

A function f : X → Y between two convergence spaces is continuous pro-
vided that f→F → x whenever F → x. Here, f→F denotes the filter on Y
generated by {f(F): F ∈ F}. Given G ∈ F(Y ), we use f←G to denote the filter
on X generated by {f−1(G): G ∈ G} whenever the latter does not contain ∅.
Given two convergence structures p and q on a set X, we say that q is finer
than p, denoted q ≥ p, whenever the identity mapping idX : (X, q) → (X, p) is
continuous.

Let CONV denote the category of convergence spaces and continuous maps
and let X be an object in CONV. The closure and interior of a subset A of
X are defined as follows:

cl A = {x ∈ X : F → x for some F and A ∈ F}

int A = {x ∈ A: F → x implies that A ∈ F}

The operators cl and int are in general not idempotent. The neighborhood
filter of x is defined by

U(x) = {V ⊆ X : x ∈ int(V )}

and A ⊆ X is open whenever int(A) = A. The convergence structure q on X
is called a pretopology on X if U(x) → x for each x ∈ X. A pretopology on
X is said to be a topology whenever U(x) has a base of open sets for each
x ∈ X. We say X is Hausdorff provided that each filter converges to at most
one point, and regular if cl F → x whenever F → x, where cl F denotes the
filter on X generated by {cl F : F ∈ F}. A Hausdorff regular convergence space
is called T3. A point x ∈ X is an adherent point of F ∈ F(X) whenever
there exists a G ≥ F such that G → x; adh F denotes the set of all adherent
points of F. We say X is compact provided that adh F 6= ∅ for each F ∈ F(X)
or equivalently if each ultrafilter on X converges. A convergence space Y is
said to be a compactification of X in CONV if Y is compact Hausdorff and

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Convergence S-compactifications

if there is a dense embedding of X into Y . Observe that compactifications are
required to be Hausdorff. A compactification Y is called regular whenever Y
is regular. If Y and Z are two compactifications of X in CONV, define Y ≥ Z
to mean that there exists a continuous map h: Y → Z such that h ◦ f = g,
where f is the dense embedding of X into Y and g is the dense embedding of
X into Z. Note that ≥ is a partial order on the set of compactifications of X
if we agree not to distinguish between isomorphic objects in CONV.

Definition 1.2. The triple (S, ·, p) is said to be a convergence monoid if it
satisfies:

(CM1) (S, ·) is a commutative monoid with identity e.
(CM2) (S, p) is a convergence space.
(CM3) The binary operation (x, y) 7→ x · y of S is continuous.

Let CM denote the category of convergence monoids and continuous homo-
morphisms. To simplify notation, we will write S for the object (S, ·, p) in
CM. Let X be a convergence space, let S be a convergence monoid and let
λ: X × S → X. Consider the following conditions on λ:

(A1) λ(x, e) = x for all x ∈ X.
(A2) λ(λ(x, s), t) = λ(x, s · t) for all x ∈ X and all s, t ∈ S.
(A3) λ is continuous.

If λ satisfies (A1) and (A2), then λ is an action of S on X, and if in addition
it satisfies (A3), we say that λ is a continuous action of S on X.

Definition 1.3. Let CA be the category whose objects consist of all triples
(X, S, λ), where X is a convergence space, S is a convergence monoid and λ
is a continuous action of S on X, and whose morphisms are pairs (f, k) of
functions of the form (X, S, λ) → (Y, T, µ) such that:

(C1) f : X → Y is a morphism in CONV,
(C2) k : S → T is a morphism in CM and
(C3) µ ◦ (f × k) = f ◦ λ.

Definition 1.4. A compactification (regular compactification) of an ob-
ject (X, S, λ) in CA is an object (Y, S, µ) in CA such that:

(COM1) Y is a compact Hausdorff (compact T3) convergence space,
(COM2) X is densely embedded in Y , where the dense embedding f is such

that
(COM3) (f, idS) is a morphism in CA.

Remark 1.5. Throughout the remainder of this work, S will always denote a
convergence monoid, X a convergence space and λ a continuous action of S on
X. We will always write p for the convergence structure on S and q for the
convergence structure on X. An object in CA of the form (Y, S, µ) is called
an S-space and for notational convenience will be denoted as (Y, µ) or Y . A
morphism (f, idS) between two S-spaces in CA will be written more simply as
f. Also, any compactification of X in CA will be called an S-compactification
of X.

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B. Losert and G. Richardson

Assume that Y and Z are two S-compactifications of X in CA with dense
embeddings f : X → Y and g : X → Z. Define Y ≥ Z to mean that there
exists a morphism h: Y → Z in CA such that h ◦ f = g. We say Y and Z are
equivalent S-compactifications of X if Y ≥ Z and Z ≥ Y . Verification of
the following lemma is straightforward and omitted here.

Lemma 1.6.

(i) The relation ≥ between S-compactifications of X defined above is a
partial order on the set of all S-compactifications of X if we agree not
to distinguish between equivalent S-compactifications.

(ii) Suppose that Y and Z are S-compactifications of X satisfying Y ≥ Z.
Then the following diagram commutes:

Y Z

X

h

f

g

and (a) h(Y − f(X)) = Z − g(X) and (b) f(X) is open in Y if and
only if g(X) is open in Z.

Definition 1.7.

(i) We call X adherence-restrictive if for each F ∈ F(X) and each
convergent filter G ∈ F(S), adh F = ∅ implies adh λ→(F × G) = ∅.

(ii) Let Y be an S-compactification of X, let f : X → Y be the dense
embedding and let µ be the action of S on Y . We say that Y is
remainder-invariant provided that µ((Y − f(X)) × S) ⊆ Y − f(X).

2. S-compactifications

Recall from Section 1 that a compactification must be Hausdorff. Unlike the
topological context, a one-point compactification in CONV is not necessarily
unique up to homeomorphism. The following one-point compactification de-
fined below is used. Pick an ω 6∈ X, let X∗ = X ∪ {ω} and let j : X → X∗ be
the natural injection. Define convergence in X∗ as follows:

H ∈ F(X∗) converges to j(x) ⇔ H ≥ j
→
F for some F ∈ F(X) that converges to x

H ∈ F(X∗) converges to ω ⇔ H ≥ j
→
F ∩ ω̇ for some F ∈ F(X) with adh F = ∅

One can check that X∗ with the above convergence is a compactification of X
in CONV provided that X is non-compact and Hausdorff. Moreover, Y ≥ X∗
in CONV for any other one-point compactification Y of X.

Let η denote the set of all ultrafilters on X which fail to converge.

Theorem 2.1. Suppose X is not compact and let Y be an S-compactification
of X. Then X is adherence-restrictive if and only if Y is remainder-invariant.

Proof. A contrapositive argument is used in each direction. Let f : X → Y be
the dense embedding, let µ be the action of S on Y and suppose that Y fails to
be remainder-invariant. Then µ(y, s) = f(x) for some y ∈ Y −f(X), s ∈ S and

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Convergence S-compactifications

x ∈ X. Let F ∈ η such that f→F → y. Then adh F = ∅ and f→(λ→(F × ṡ)) =
(f ◦ λ)

→

(F × ṡ) = (µ ◦ (f × idS))
→

(F × ṡ) = µ→(f→F × ṡ) → µ(y, s) = f(x).
Since f is an embedding, λ→(F×ṡ) → x and thus adh(λ→(F×ṡ)) 6= ∅, proving
that X is not adherence-restrictive.

Conversely, suppose that X is not adherence-restrictive. Then there exists

F ∈ F(X) with adh F = ∅ and x ∈ adh(λ→(F×G)) for some G
p
→ s and x ∈ X.

Let H be an ultrafilter on X such that H → x and H ≥ λ→(F × G). Let K be
an ultrafilter such that K ≥ F×G and H = λ→K. Note that the projection KX
of K onto X is finer than F. Since Y is compact and adh F = ∅, f→KX → y
for some y ∈ Y − f(X), hence f→H = f→(λ→K) ≥ (f ◦ λ)

→

(KX × G) =
(µ ◦ (f × idS))

→
(KX × G) = µ

→(f→KX × G) → µ(y, s). However, H → x
implies that f→H → f(x), and thus µ(y, s) = f(x). Hence Y fails to be
remainder-invariant. �

Theorem 2.2. If X is non-compact and Hausdorff, then it has a one-point re-
mainder-invariant S-compactification if and only if X is adherence-restrictive.

Proof. Assume that X is adherence-restrictive and let X∗ be the compactifica-
tion of X in CONV discussed above. Define λ∗ : X∗ × S → X∗ by

λ∗(y, s) =

{

j(λ(y, s)), if y ∈ j(X)

ω, if y = ω.

Some simple algebra reveals that λ∗ is an action of S on X∗. It is shown that
λ∗ is continuous. Suppose that H ∈ F(X∗) converges to j(x) and G ∈ F(X)
converges to s. Then there exists F ∈ F(X) that converges to x such that
H ≥ j→F. It follows from the definition of λ∗ and the continuity of λ that
λ∗
→
(j→F × G) = (j ◦ λ)

→

(F × G) → j(λ(x, s)). Next, suppose that H ∈ F(X∗)
converges to ω and G ∈ F(X) converges to s. Then there exists F ∈ F(X)
such that adh F = ∅ and H ≥ j→F ∩ ω̇. Since X is adherence-restrictive,
adh(λ→(F × G)) = ∅. It follows that λ→

∗
(H × G) ≥ λ→

∗
((j→F ∩ ω̇) × G) =

λ→
∗
(j→F × G) ∩ λ→

∗
(ω̇ × G) = j→(λ→(F × G)) ∩ ω̇ → ω. This proves that λ∗ is

a continuous action of S on X∗. Moreover, λ∗ ◦ (j × idS) = j ◦ λ and thus X∗
(with λ∗ as the action) is an S-compactification of X. The compactification is
also remainder-invariant by construction. The converse follows from Theorem
2.1. �

Theorem 2.3. Assume that X is non-compact and Hausdorff. Then it has
a smallest remainder-invariant S-compactification if and only if X is adher-
ence-restrictive and the image of X is open in each of its remainder-invariant
S-compactifications. Moreover, if there is a smallest remainder-invariant S-
compactification, it is equivalent to the one-point S-compactification X∗.

Proof. Suppose that X has a smallest remainder-invariant S-compactifiction.
According to Theorem 2.1, X is adherence-restrictive, and thus by Theorem
2.2, X∗ is a remainder-invariant S-compactification of X. Applying Lemma

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B. Losert and G. Richardson

1.6 (ii), it follows that X∗ is equivalent to the smallest remainder-invariant S-
compactification. Moreover, part (b) of Lemma 1.6 (ii) implies that the image
of X is open in each of its remainder-invariant S-compactifications.

Conversely, suppose that X is adherence-restrictive and that the image of
X is open in each of its remainder-invariant S-compactifications. According
to Theorem 2.2, X∗ is a one-point S-compactification of X and j(X) is open
in X∗. Let Y be any remainder-invariant S-compactication of X having dense
embedding f and action µ. Define h: Y → X∗ by h(f(x)) = j(x) and h(y) = ω
for each x ∈ X and y ∈ Y − f(X). We show that h is continuous: If a filter
H on Y converges to f(x), then f(X) ∈ H since f(X) is open in Y , and
thus F := f←H → x and h→H = (h ◦ f)

→
F = j→F → j(x) = h(f(x)). Next,

assume that H → y ∈ Y −f(X). If Y −f(X) ∈ H, then h→H = ω̇ → ω = h(y).
Otherwise, F = f←H exists, adh F = ∅ and h→H ≥ j→F ∩ ω̇ → ω.

Next, we show λ∗ ◦ (h × idS) = h ◦ µ: If x ∈ X and s ∈ S, then (λ∗ ◦
(h × idS))(f(x), s) = λ∗(h(f(x)), s) = j(λ(x, s)) = (h ◦ f)(λ(x, s)) = (h ◦
(f ◦ λ))(x, s) = h(µ ◦ (f × idS))(x, s) = (h ◦ µ)(f(x), s), and thus (λ∗ ◦ (h ×
idS))(f(x), s) = (h ◦ µ)(f(x), s). Finally, if y ∈ Y − f(X), then (λ∗(h ◦
idS))(y, s) = λ∗(h(y), s) = λ∗(ω, s) = ω = (h ◦ µ)(y, s). The last equality
is valid since Y is remainder-invariant. Therefore, λ∗ ◦ (h × idS) = h ◦ µ, and
thus h is a morphism in CA. This concludes the proof that X∗ is the smallest
remainder-invariant S-compactification of X. �

We say that S is a convergence group whenever S is a group and the
function that maps elements of S to their inverses is continuous. In what
follows, we will write s−1 for the inverse of s ∈ S. Also, given a filter G on S,
we will write G−1 for the filter on S generated by {G−1 : G ∈ G}, where G−1

denotes the set of inverses of the elements of G. The next result should be
contrasted with Theorem 2.1.

Theorem 2.4. If S is a convergence group, then X is adherence-restrictive.

Proof. Suppose X fails to be adherence-restrictive. Then there exists F ∈
F(X), G ∈ F(S), s ∈ S and x ∈ X such that adh F = ∅ and G → s and
x ∈ adh(λ→(F × G)). Let H be an ultrafilter such that H ≥ λ→(F × G) and
H → x. Choose an ultrafilter L ≥ F × G for which H = λ→L. Since S is
a convergence group, G−1 → s−1, and thus λ→(H × G−1) → λ(x, s−1). It is
shown that λ→(λ→L × G−1) ∨ LX exists, LX being the projection of L on X.
Let L ∈ L, let LX denote the projection of L on X and let G ∈ G. Choose any
F ∈ F. Since L ≥ F × G, L ∩ (F × G) 6= ∅. Let (y, t) ∈ L ∩ (F × G). Then
(λ(y, t), t−1) ∈ λ(L) × G−1 and thus y ∈ λ(λ(L) × G−1) ∩ LX, which means
λ→(λ→L × G−1) ∨ LX exists. Since λ

→(H × G−1) → λ(x, s−1), we have that
F ≤ LX ≤ λ

→(λ→L × G−1) ∨ LX = λ
→(H→ × G−1) ∨ LX → λ(x, s

−1), which
contradicts adh F = ∅. �

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Convergence S-compactifications

The next result appears as Theorem 3.2 [9].

Theorem 2.5 ([9]). If X is non-compact and Hausdorff, then the following
are equivalent:

(i) X has a smallest compactification in CONV.
(ii) The image of X is open in each of its compactifications.
(iii) The set η of non-convergent ultrafilters on X is finite.
(iv) X has a largest compactification in CONV.

Example 2.6. Suppose S = {e} is the trivial group, X Hausdorff and η
is infinite. Define λ: X × S → X by λ(x, s) = x. Then λ is a continuous
action and thus X is an S-space. Let Y be any compactification of X with
dense embedding f. Define µ: Y × S → S by µ(y, s) = y. Then µ is a
continuous action of S on Y and µ ◦ (f × idS) = f ◦ λ. It follows that Y
is an S-compactification of X. In this case, there is a bijection between the
compactifications of X and the S-compactifications of X. Since η is infinite, it
follows from Theorem 2.5 that there fails to exists either a smallest or a largest
S-compactification of X.

Example 2.7. Let X := [0, 1) and S := (0, 1] be equipped with the usual
topologies and let the operation on S be multiplication. Define λ: X × S → X
by λ(x, s) = xs. Then λ is a continuous action and thus X is an S-space. Let
X∗ = [0, 1] have the usual topology (note that ω = 1 in this case). Define
µ: X∗ × S → X∗ by µ(y, s) = ys. Then µ is a continuous action of S on X∗
and the S-space (X∗, µ) is an S-compactification of X. Since µ(ω, s) = s 6= ω
whenever s 6= 1, (X∗, µ) is not remainder-invariant. Since the action λ∗ of S
on X∗ fails to be continuous at each (ω, s), s 6= 1, the S-space (X∗, λ∗) is not
an S-compactification of X. Ergo, X is not adherence-restrictive.

3. Regular S-compactifications

Recall that U(x) denotes the neighborhood filter of x ∈ X, and that the

convergence structure q on X is a pretopology provided that U(x)
q
→ x for

each x ∈ X. If q is not a pretopology, define F
πq
→ x if and only if F ≥ U(x).

Then πq is a pretopology on X and πX := (X, πq) is called the pretopological
modification of X.

Theorem 3.1 ([6]). The convergence space X has a regular compactification
in CONV if and only if

(i) X and πX agree on convergence of ultrafilters, and
(ii) πX is a completely regular topological space.

We say X is completely regular provided that it is T3 and agrees on ul-
trafilter convergence with a completely regular topological space. With this
definition, we can restate Theorem 3.1 above as: X has a regular compactifi-
cation in CONV if and only if X is a completely regular convergence space.

Recall that an object Y in CA is a regular S-compactification of X if Y is
a regular compactification of X and if µ◦(f ×idS) = f ◦λ, where f : X → Y is

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B. Losert and G. Richardson

the dense embedding. A convergence space is locally compact provided that
each convergent filter contains a compact subset. According to Proposition
2.3 [8], a convergence space is locally compact if and only if each convergent
ultrafilter contains a compact subset.

Theorem 3.2. If X is adherence-restrictive and completely regular, then the
statements (i) – (iii) are equivalent and (iii) implies (iv), where

(i) X has a one-point regular remainder-invariant S-compactification.
(ii) X is locally compact.
(iii) X has a regular remainder-invariant S-compactification and the image

of X is open in each such compactification.
(iv) X has a smallest regular remainder-invariant S-compactification.

Proof. (i) implies (ii): Let Y be a one-point regular S-compactification of
X and let f : X → Y be the dense embedding. According to Theorem 3.1,
f : πX → πY is a dense embedding in the category TOP of topological spaces
and thus πY is a T3-compactification of πX in TOP, which means πX is locally
compact. Again, by Theorem 3.1, X and πX agree on ultrafilter convergence
so X is also locally compact.

(ii) implies (i): Let X∗ be the one-point S-compactification defined in Section
2 with dense embedding j : X → X∗. It must be shown that X∗ is regular.
Assume that H is a filter on X∗ that converges to j(x). Then H ≥ j

→F for
some F ∈ F(X) that converges to x. Since X is completely regular and locally
compact, cl F → x and F contains a compact subset A of X. Observe that
ω 6∈ cl j(A): otherwise, if there is an ultrafilter K on X∗ that contains j(A)
and converges to ω, then j←K is a filter on X that contains A and has empty
adherence, contradicting that A is compact. It follows that X∗ − j(A) ∈ K for
each ultrafilter K on X∗ that converges to ω. Hence cl j(A) = j(cl A) = j(A)
and thus cl H ≥ cl j→F = j→(cl F) → j(x). Moreover, if H → ω, then H ≥
j→F ∩ ω̇ for some F ∈ F(X) such that adh(F) = ∅. Since X is completely
regular, it has the same ultrafilter convergence as πX, hence adh(cl F) = ∅ and
thus cl H ≥ cl(j→F ∩ ω̇) = j→(cl F) ∩ ω̇ → ω. This proves that X∗ is regular
and thus X∗ is a one-point regular remainder-invariant S-compactification of
X.

(ii) implies (iii): Since (ii) implies (i), X has a regular remainder-invariant S-
compactification. Let Y be any regular remainder-invariant S-compactification
with dense embedding f : X → Y . We now show that f(X) is open in Y . Since
f : πX → πY is a dense embedding in TOP, πY is a Hausdorff compactification
of πX in TOP. Since X and πX agree on ultrafilter convergence, πX and X
are locally compact. Thus, f(X) is open in πY and hence open in Y .

(iii) implies (ii): Let Y be a regular S-compactification of X with dense em-
bedding f : X → Y such that f(X) is open in Y . Since πY is a compactification
of πX in TOP, πX is locally compact and thus X is locally compact.

(iii) implies (iv): Since (iii) implies (i), X∗ is a regular remainder-invariant
S-compactification. Suppose that Y is any regular remainder-invariant S-com-
pactification of X with dense embedding f : X → Y . By hypothesis, f(X) is

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Convergence S-compactifications

open in Y , so the proof of Theorem 2.3 shows that Y ≥ X∗, hence X∗ is the
smallest regular remainder-invariant S-compactification of X. �

Theorem 3.3. If X has a regular S-compactification, then it has a largest regu-
lar S-compactification, which is remainder invariant whenever X is adherence-
restrictive.

Proof. Let (Yα) be the family of all regular S-compactifications of X. For each
α, let fα : X → Yα be the dense embedding and let µα be the action of S on
Yα. Let Y =

∏

Yα be the product space in CONV and define µ: Y × S → Y
so that µ(y, s) = (µα(yα), s), where y = (yα). It is straightforward to check
that µ is an action of S on Y . Let πα : Y → Yα denote the projection onto Yα.
Since the diagram below is commutative,

Y × S Y

Yα × S Yα

µ

πα×idS πα

µα

it follows that µ is continuous. This proves that Y is an S-space as well as a
compact T3 object in CONV.

Define f : X → Y by f(x) = (fα(x)). Since fα = πα ◦ f for each α, f is con-
tinuous. Moreover, if F ∈ F(X) and f→F → f(x), then f→α F = π

→

α (f
→F) →

fα(x). Since fα is an embedding, F → x and consequently f is an embedding.
Since µ ◦ (f × idS) = f ◦ λ, f is an embedding in CA. Note that the following
diagram is commutative in CA:

Y Yα

X

πα

f fα

Finally, let Z = cl f(X) and let δ denote the restriction of µ on Z × S.
Observe that Z is an S-space since δ(Z×S) = δ(cl f(X)×S) ⊆ cl δ(f(X)×S) =
cl µ(f(X)×S) = cl(µ◦(f ×idS))(X ×S) = cl(f ◦λ)(X ×S) = cl f(λ(X ×S)) ⊆
cl f(X) = Z. It follows that Z is the largest regular S-compactification of X.
In the event that Z is adherence-restrictive, it follows from Theorem 2.1 that
Z is remainder-invariant. �

Consideration is now given to the question as to when an S-space has a
regular S-compactification. For this discussion, it is convenient to introduce
the notion of a Cauchy space.

Definition 3.4. The pair (X, C ) is called a Cauchy space and C a Cauchy
structure whenever C ⊆ F(X) obeys:

(CA1) ẋ ∈ C for each x ∈ X.
(CA2) F ≥ G ∈ C implies F ∈ C .
(CA3) F ∩ G ∈ C whenever F, G ∈ C and F ∨ G exists.

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B. Losert and G. Richardson

The elements of a Cauchy structure are called Cauchy filters. A function
f between Cauchy spaces is called Cauchy continuous if f→F is a Cauchy
filter whenever F is a Cauchy filter. We use CHY to denote the category of
Cauchy spaces and Cauchy continuous functions.

The first study of completions resembling a Cauchy space defined above
seems to be due to Keller [5]. The interested reader is referred to Lowen [10],
Preuss [11] and Reed [12] for a thorough treatment of Cauchy spaces.

If for all F, G ∈ F(X) and all x ∈ X, F → x and G → x imply F ∩ G → x,
then X is called a limit space. We use LIM to denote the full subcategory
of CONV whose objects are limit spaces. If C ⊆ F(X) is a Cauchy structure,

then qC : F(X) → P(X) defined by F
qC
→ x if and only if F ∩ ẋ ∈ C is a

convergence structure making (X, qC ) a limit space. If f : (X, C ) → (Y, D) is
Cauchy continuous, then f : (X, qC ) → (Y, qD ) is continuous, but the converse
is in general false. If X is a Hausdorff limit space, then C = {F ∈ F(X): F
converges} is a Cauchy structure and qC = q.

A compactification Y of X in CONV with dense embedding f is said to be
strict provided that whenever H ∈ F(Y ) converges to some y ∈ Y , there exists
an F ∈ F(X) such that f→F → y and H ≥ cl f→F. All our previous definitions
(S-compactifications, etc.) apply to the objects in LIM.

Theorem 3.5. Suppose both X and S are Hausdorff limit spaces. Let Y be
any strict regular compactification of X in LIM with dense embedding f, let
C = {F ∈ F(X): f→F converges} and let S = {G ∈ F(S): G converges}.
Then there exists a continuous action µ of S on Y making Y into a regular
S-compactification of X if and only if λ: (X, C ) × (S, S ) → (X, C ) is Cauchy
continuous.

Proof. Suppose there exists a continuous action µ of S on Y making Y into
a regular S-compactification of X. Then C is a Cauchy structure. We now
show that λ: (X, C ) × (S, S ) → (X, C ) is Cauchy continuous. Let F ∈ C and
G ∈ S . Then f→F → y and G → s for some y ∈ Y and s ∈ S. It follows that
f→(λ→(F × G)) = (µ ◦ (f × idS))

→

(F × G) = µ→(f→F × G) → µ(y, s), which
means λ→(F × G) ∈ C , which means λ is Cauchy continuous.

Conversely, suppose that λ is Cauchy continuous. Define µ: Y × S → Y by

µ(y, s) =

{

f(λ(x, s)), if y = f(x) for some x ∈ X,

lim (f ◦ λ)
→

(F × G), where f→F → y ∈ Y − f(X) and G → s.

Note that the above is well-defined. Indeed, if F1 and F2 are filters on X such
that f→F1 and f

→F2 converge to y and G1 and G2 are filters on S that converge
to s, then f→(F1∩F2) → y and G1∩G2 → s since Y and S are limit spaces, hence
F1 ∩ F2 ∈ C . Moreover, since λ is Cauchy continuous, λ

→(F1 × G1) ∩ λ
→(F2 ×

G2) ≥ λ
→((F1 ∩ F2) × (G1 ∩ G2)) ∈ C . Thus, f

→(λ→(F1 × G1) ∩ λ
→(F2 × G2))

converges and lim (f ◦ λ)
→
(F1 × G1) = lim (f ◦ λ)

→
(F2 × G2) in Y .

We now show that µ is an action. First, we have that µ(f(x), e) = f(λ(x, e)) =
f(x) and µ(y, e) = lim (f ◦ λ)

→

(F × ė) = lim f→(λ→(F × ė)) = y, where
f→F → y ∈ Y − f(X). Second, if x ∈ X and s, t ∈ S, then µ(µ(f(x), s), t) =

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Convergence S-compactifications

µ(f(λ(x, s)), t) = f(λ(λ(x, s), t)) = f(λ(x, s · t)) = µ(f(x), s · t). Third, sup-
pose y ∈ Y − f(X) and f→F → y and let s, t ∈ S. If µ(y, s) = f(x) for some
x ∈ X, then since f is an embedding and f(x) = µ(y, s) = lim (f ◦ λ)

→
(F ×

ṡ) = lim f→(λ→(F × ṡ)), we have that λ→(F × ṡ) → x and so µ(y, s · t) =
lim (f ◦ λ)

→

(F× ˙s · t) = lim f→(λ→(λ→(F×ṡ)×ṫ)) = f(λ(x, t)) = µ(f(x), t) =
µ(µ(y, s), t). Otherwise, if µ(y, s) ∈ Y − f(X), then since f→(λ→(F × ṡ)) →
µ(y, s) we have that µ(µ(y, s), t) = lim (f ◦ λ)

→
(λ→(F×ṡ)×ṫ) = lim (f ◦ λ)

→
(F×

˙s · t).
We now show that µ is continuous. First, we prove that if A ⊆ X and

G ⊆ S, then µ(cl f(A) × G) ⊆ cl(f ◦ λ)(A × G): Let (y, s) ∈ cl f(A) × G.
If y = f(x), then x ∈ cl A, and since (f ◦ λ)(cl A × G) ⊆ cl(f ◦ λ)(A × G),
µ(f(x), s) = f(λ(x, s)) ∈ cl(f ◦ λ)(A × G). Otherwise, if y ∈ Y − f(X) and
f→F → y with A ∈ F, then A × G ∈ F × ṡ, (f ◦ λ)

→

(F × ṡ) → µ(y, s),
hence µ(y, s) ∈ cl(f ◦ λ)(A × G), hence µ(cl f(A) × G) ⊆ cl(f ◦ λ(A × G)) as
claimed. Now suppose H is a filter on Y and that G is a filter on S converging
to s. If H → f(x), then since Y is a strict regular compactification of X in
LIM, there exists an F ∈ F(X) such that f→F → y and H ≥ cl f→F, hence
λ→(F×G) → λ(x, s) and µ→(H×G) ≥ µ→(cl f→F×G) ≥ cl (f ◦ λ)

→
(F×G) →

f(λ(x, s)) = µ(f(x), s). If H → y ∈ Y − f(X), then since Y is strict, there is
an F ∈ F(X) that converges to x such that H ≥ cl f→F, hence µ→(H × G) ≥
µ→(cl f→F × G) ≥ cl (f ◦ λ)

→

(F × G), hence by the regularity of Y and the
fact that (f ◦ λ)

→

(F × G) → µ(y, s) we have that µ→(H × G) → µ(y, s). This
proves that µ is continuous, and since µ ◦ (f × idS) = f ◦ λ, Y is a regular
S-compactification of X. �

Recall from Theorem 3.1 that a T3 convergence space has a regular com-
pactification in CONV if and only if it is compeletely regular. This result is
also valid in the subcategory LIM. The following additional result is proved in
Theorem 2 [6].

Theorem 3.6. If X is a completely regular convergence (limit) space, then it
has a largest regular compactification in CONV (respectively, LIM). The largest
regular compactification is strict.

The largest regular compactification guaranteed by Theorem 3.6 is called the
Stone-Čech compactification for convergence (limit) spaces. Combining the
two previous theorems yields the next result.

Theorem 3.7. Suppose X is a completely regular limit space with Stone-Čech
compactification βX in LIM and dense embedding f : X → βX and suppose S
is a Hausdorff limit space. Let S = {G ∈ F(S): G converges} and C = {F ∈
F(X): f→F converges}. Then λ can be extended to a continuous action of S
on βX making it into a regular S-compactification of X if and only if λ is
Cauchy continuous. Moreover, βX is the largest regular S-compactification of
X whenever λ is Cauchy continuous.

Proof. Let µ be the action of S on βX. According to Theorem 3.6, βX is
strict, so by Theorem 3.5, βX is a regular S-compactification of X if and only

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 131



B. Losert and G. Richardson

if λ is Cauchy continuous. Suppose λ is Cauchy continuous. Then f is a
dense embedding in CA and µ ◦ (f × idS) = f ◦ λ. It remains to show that
βX is the largest regular S-compactification of X. Let Y be any regular S-
compactification of X with dense embedding g and action δ. Then δ ◦ (g ×
idS) = g ◦ λ. Since βX is the Stone-Čech compactification of X in LIM, there
exists a continuous function h: βX → Y such that h ◦ f = g. We show that
δ◦(h×idS) = h◦µ on f(X): If x ∈ X and s ∈ S, then (δ◦(h×idS))(f(x), s) =
δ((h ◦ f)(x), s) = δ(g(x), s) = (δ ◦ (g × idS))(x, s) = (g ◦ λ)(x, s) = (h ◦ f ◦
λ)(x, s) = (h◦ (f ◦ λ))(x, s) = (h◦ (µ◦ (f × idS)))(x, s) = (h◦ µ)(f(x), s). Since
f(X) is dense in βX, δ ◦ (h × idS) = h ◦ µ on βX and thus h is a morphism in
CA. This means βX ≥ Y , hence βX is the largest regular S-compactification
of X. �

4. Generalized Quotient Spaces

Let GQ denote the full subcategory of CA whose objects (Y, T, µ) satisfy:

(i) T is a commutative.
(ii) µ(·, t) is injective for each fixed t ∈ T .

Suppose X is an object in GQ. Define a relation ∽ on X × S so that (x, s) ∽
(y, t) if and only if λ(x, t) = (y, s). One can readily check that ∽ is an equiv-
alence relation. We use 〈x, s〉 for the equivalence class containing (x, s) and
θ : X × S → (X × S)/ ∽ for the quotient map. We call (X × S)/ ∽ the
generalized quotient of X and we denote it by B(X).

Define Λ: (X × S) × S → (X × S) by Λ((x, s), t) = (λ(x, t), s) and define
λB : B(X) × S → B(X) by λB(〈x, s〉 , t) = 〈λ(x, t), s〉. It is shown in Theorem
2.4 (b) of [2] that Λ and λB are continuous actions, which means that X × S
and B(X) are S-spaces. In fact, the following diagram is commutative:

(X × S) × S X × S

B(X) × S B(X)

Λ

(θ,idS) θ

λB

Theorem 4.1. Suppose X is an object in GQ. If S is compact and X is not
compact and Hausdorff, then B(X∗) is a one-point S-compactification of B(X),
where X∗ is the one-point S-compactification of X given in Thereom 2.2.

Proof. First we show that λ∗
B
: B(X)×S → B(X) defined by λ∗

B
(〈j(x), s〉 , t) =

〈(j ◦ λ)(x, t), s〉 and λ∗
B
(〈ω, s〉 , t) = 〈λ∗(ω, t), s〉 = 〈ω, s〉 is an action. Note that

λ∗
B
(〈j(x), s〉 , e) = 〈j ◦ λ(x, e), s〉 = 〈j(x), s〉 and λ∗

B
(〈ω, s〉 , e) = 〈ω, s〉. Also,

λ∗
B
(λ∗

B
(〈j(x), t〉 , s), u) = λ∗

B
(〈(j ◦ λ)(x, t), s〉 , u)

= 〈(j ◦ λ)(λ(x, t), u), s〉

= 〈(j ◦ λ)(x, t · u), s〉

= λ∗
B
(〈j(x), s〉 , tu)

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Convergence S-compactifications

and

λ∗
B
(λ∗

B
(〈ω, s〉 , t), u) = λ∗

B
(〈λ∗(ω, t), s〉 , u)

= λ∗
B
(〈ω, s〉 , u)

= 〈λ∗(ω, u), s〉

= 〈ω, s〉 = λ∗
B
(〈ω, s〉 , t · u)

This proves that λB is an action.
Define Λ∗ : (X∗×S)×S → X∗×S so that Λ

∗((j(x), s), t) = (λ∗(j(x), t), s) =
((j ◦ λ)(x, t), s) and Λ∗((ω, s), t) = (λ∗(ω, t), s) = (ω, s). Note that Λ

∗ is an
action of S on X∗ and it is continuous since it is the composition:

((z, s), t) 7→ ((z, t), s) 7→ (λ∗(z, t), s)

It follows that X∗ × S is an object in GQ. Consider the following commutative
diagram:

(X∗ × S) × S X∗ × S

B(X∗) × S B(X∗)

Λ∗

(θ∗,idS) θ

λ
∗

B

where θ∗ : X∗ × S → B(X∗) is the quotient map in CONV. Since (θ
∗, idS) is a

quotient map in CONV, the diagram above shows that λ∗
B
is continuous, which

means that B(X∗) is an S-space.
Since 〈x, s〉 ∈ B(X) if and only if 〈j(x), s〉 ∈ B(X∗) and since 〈ω, s〉 =

{(ω, t): t ∈ S}, it follows that γ : B(X) → B(X∗) defined by γ(〈x, s〉) =
〈j(x), s〉 is an injection and B(X∗) − γ(B(X)) is a singleton set containing
〈ω, s〉. We now proceed to show that γ is a dense embedding.

First, observe that the diagram below is commutative:

X × S B(X)

X∗ × S B(X∗)

θ

(j,idS) γ

θ
∗

Moreover, since θ is a quotient map and γ ◦ θ = θ∗ ◦ (j, idS) is continuous
in CONV, it follows that γ is continuous. Next, assume that H ∈ F(B(X))
such that γ→H → 〈j(x), s〉. Then there exists (j(x′), s′) ∽ (j(x), s) and K →
(j(x′), s′) such that θ∗→K = γ→H. The product convergence structure on
X∗×S implies that for some F ∈ F(X) and some G ∈ F(S), F → x

′, G → s′ and
K ≥ j→F ×G, hence (γ ◦ θ)

→

(F ×G) = (θ∗ ◦ (j × idS))
→

(F ×G) = θ∗
→
(j→F ×

G) ≤ θ∗→K = γ→H, hence L := θ→(F × G) → 〈x′, s′〉 and γ→L ≤ γ→H, hence
L ≤ H since γ is an injection. It follows that H → 〈x′, s′〉 = 〈x, s〉, proving
that γ is an embedding.

For any fixed s ∈ S, B(X∗) − γ(B(X)) = {〈ω, s〉}. Since X is not compact,
there exists an ultrafilter F ∈ F(X) that fails to converge, hence j→F → ω,
j→F×ṡ → (ω, s) and θ∗→(j→F×ṡ) → 〈ω, s〉. It follows that γ→(θ→(F×G)) =

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B. Losert and G. Richardson

(θ∗ ◦ (j, idS))
→

(F × G) = θ∗→(j→F × G) → 〈ω, s〉, and since θ→(F × G) ∈
F(B(X)), we conclude that γ is a dense embedding. Moreover, since X∗ and
S are compact, B(X∗) is a one-point S-compactification of B(X) in CONV.

We now verify that γ is a morphism in CA: Since (λ∗
B
◦(γ ×idS))(〈x, s〉 , t) =

λ∗
B
(〈j(x), s〉 , t) = 〈(j ◦ λ)(x, t), s〉 = γ(〈λ(x, t), s〉) = (γ ◦ λB)(〈x, s〉 , t), we

have that λ∗
B
◦ (γ × idS) = γ ◦ λB. In conclusion, B(X∗) is a one-point S-

compactification of B(X). �

A regular S-compactification of a generalized quotient S-space is given be-
low. The following definition is needed: A continuous surjection f : X → Y
between convergence spaces is said to be proper if for each ultrafilter F on X,
f→F → y implies F → x for some x ∈ f−1({y}). It is shown in Proposition
3.2 in [7] that proper maps preserve closures. Observe that if f : X → Y is a
continuous surjection and X is compact and Y is Hausdorff, then f is a proper
map.

Theorem 4.2. Suppose X is an adherence-restrictive object in GQ and let Y
be a strict regular S-compactification of X with dense embedding f and action
µ. Assume that S is compact and regular and define h: B(X) → B(Y ) by
h(〈x, s〉) = 〈f(x), s〉. Then B(Y ) is a regular S-compactification of B(X) with
dense embedding h.

Proof. Consider the following commutative diagram:

X × S B(X)

Y × S B(Y )

θ

(f,idS) h

θY

Note that h is continuous since θ is a quotient map in CONV and h◦θ = θY ◦(f×
idS) is continuous. We now prove that h is an injection: Suppose that 〈x, s〉 6=
〈z, t〉. Then λ(x, t) 6= λ(z, s), h(〈x, s〉) = 〈f(x), s〉 and h(〈z, t〉) = 〈f(z), t〉.
However, f is injective, so µ(f(x), t) = f(λ(x, t)) 6= f(λ(z, s)) = µ(f(z), s). It
follows that h(〈x, s〉) = 〈f(x), s〉 6= 〈f(z), t〉 = h(〈z, t〉), proving that h is an
injection as claimed.

Next, suppose that H is a filter on B(X) such that h→H → h(〈x, s〉) =
〈f(x), s〉. We show that H → 〈x, s〉: The quotient convergence structure on
B(Y ) implies that there is a filter K on Y that converges to y and a filter G
on S that converges to t such that h→H ≥ θ→Y (K × G) and 〈y, t〉 = 〈f(x), s〉.
Since Y is a strict regular compactification of X in CONV, there exists an
F ∈ F(X) such that f→F → y and K ≥ cl f→F. It follows that h→H ≥ θ→Y (K×
G) ≥ θ→Y (cl f

→F × G), hence H ≥ h←(θ→Y (cl f
→F × G)). We now show that

h←(θ→Y (cl f
→F × G)) ≥ θ→(cl F × G) by verifying that h−1(θY (cl f(F) × G)) ⊆

θ(cl F × G) for arbitrary F ∈ F and G ∈ G. Let 〈v, a〉 ∈ h−1(θY (cl f(F) × G)).
Then h(〈v, a〉) = 〈f(v), a〉 ∈ θY (cl f(F) × G) implies that (w, b) ∈ cl f(F) × G
for some 〈w, b〉 = 〈f(v), a〉, hence µ(w, a) = µ(f(v), b) = f(λ(v, b)) ∈ f(X).
However, by Theorem 2.1, Y is remainder-invariant and thus w = f(z) ∈

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Convergence S-compactifications

cl f(F) for some z ∈ X. Since f is an embedding, it follows that z ∈ cl F .
Moreover, µ(w, a) = µ(f(z), a) = f(λ(z, a)) = f(λ(v, b)) and so λ(z, a) =
λ(v, b) because f is an injection. Hence 〈v, a〉 = 〈z, b〉 ∈ θ(z, b) ∈ θ(cl F × G)
shows that h−1(θY (cl f(F)×G)) ⊆ θ(cl F ×G), and thus h

←(θ→Y (cl f
→F×G)) ≥

θ→(cl F × G) as claimed.
Recall K is a filter on Y that converges to y and that G is a filter on S that

converges to t where 〈y, t〉 = 〈f(x), s〉. Again, since Y is remainder-invariant,
µ(y, s) = µ(f(x), t) = f(λ(x, t)) ∈ f(X) and so y = f(x′) for some x′ ∈ X.
Thus, µ(y, s) = µ(f(x′), s) = f(λ(x′, s)) = f(λ(x, t)) and so λ(x, t) = λ(x′, s),
which means 〈x, s〉 = 〈x′, t〉. Since f→F → y = f(x′), F → x′ and X is regular,
we have that cl F → x′, hence H ≥ θ→(cl F × G) → θ(x′, t) = 〈x′, t〉 = 〈x, s〉,
hence h is an embedding. Since S is compact, it follows that Y × S is compact,
proving that B(Y ) is compact.

Let 〈y, s〉 ∈ B(Y ). Then there exists F ∈ F(X) such that f→F → y, hence
(h ◦ θ)

→

(F × ṡ) = (θY ◦ (f × idS))
→

(F × ṡ) = θ→Y (f
→F × ṡ) → θY (y, s) =

〈(y, s)〉, which proves that h is a dense embedding. Since Y is Hausdorff, it
follows from Theorem 4.1 in [1] that B(Y ) is also Hausdorff. In conclusion,
B(Y ) is a compactification of B(X) in CONV, and by employing Theorem 2.4
(b) in [2], B(Y ) is an S-compactification of B(X).

We now show that B(Y ) is regular: Suppose that H is a filter on B(Y )
that converges to 〈y, s〉. Then there exists a filter K on Y that converges to
some y′ and a filter G on S that converges to s′ such that H ≥ θ→Y (K × G) and
〈y, s〉 = 〈y′, s′〉. Since Y and S are compact and B(Y ) is Hausdorff, it follows
from Proposition 3.2 in [7] that θY is a proper map and thus closure preserving.
This means that cl H ≥ cl θ→Y (K × G) = θ

→

Y (cl K × cl G) → θY (y
′, s′) = 〈y, s〉

since Y and S are regular. Therefore, cl H → 〈y, s〉, proving that B(Y ) is a
regular. �

Employing Theorems 3.2, 4.1, and 4.2 gives the following result.

Corollary 4.3. Suppose X is an object in GQ that is adherence-restrictive,
locally compact and completely regular and suppose S is compact and completely
regular. Then B(X∗) is a one-point regular S-compactification of B(X), where
X∗ is the one-point regular S-compactification of X.

Let us now consider the topological case. Let τq be the finest topology on
X coarser than the convergence structure q of X and let τX := (X, τq) denote
the topological modification of X. Recall that θ : X × S → B(X) is a quotient
map in CONV. According to Theorem 4.2 in [8], if either X or S is a locally
compact Hausdorff topological space, then τ(X × S) = τX × τS and thus θ is
a quotient map in TOP. In this case, τB(X) is the generalized quotient of τX
(acted upon by τS) in TOP. The final result pertains to generalized quotient
spaces in TOP and the proof follows from the preceeding remarks along with
Theorem 4.2.

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 135



B. Losert and G. Richardson

Corollary 4.4. Suppose X is an adherence-restrictive object in GQ and that
S is a compact Hausdorff topological space. Let Y be a topological S-compacti-
fication of X. Then τB(X) and τB(Y ) are generalized quotients in TOP and
τB(Y ) is a topological S-compactification of τB(X).

5. Conclusion

In general, an S-space has neither a smallest nor a largest S-compactifica-
tion. Conditions are given for the existence of a smallest (regular) S-compact-
ification in Theorem 2.3 (respectively, Theorem 3.2). Whether or not (iv) of
Theorem 3.2 is equivalent to the other three statements listed in the theorem
is unknown to the authors. Each S-space having a regular S-compactification
has a largest such. Theorem 3.5 and Theorem 3.7 address the question as to
when a regular compactification in LIM can be used to form a regular S-com-
pactification in CA. It is shown in Theorem 4.2 that the process of forming
a regular S-compactification remains invariant under the operation of taking
“generalized quotients.”

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