@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 1, 2003

pp. 143–155

On classes of T0 spaces admitting completions

Eraldo Giuli
∗

Abstract. For a given class X of T0 spaces the existence of a sub-
class C, having the same properties that the class of complete metric
spaces has in the class of all metric spaces and non-expansive maps, is
investigated. A positive example is the class of all T0 spaces, with C
the class of sober T0 spaces, and a negative example is the class of Ty-
chonoff spaces. We prove that X has the previous property (i.e., admits
completions) whenever it is the class of T0 spaces of an hereditary core-
flective subcategory of a suitable supercategory of the category Top of
topological spaces. Two classes of examples are provided.

2000 AMS Classification: Primary: 54A05, 54B30, 54D25, 54G10, 18A30.
Secondary: 54D05, 54D10, 54D30.

Keywords: Affine set, T0, sober and injective space, compact space, com-
pletion, Zariski closure, topological category, coreflective subcategory.

1. Introduction

Let Met be the category of metric spaces and non-expansive maps and let
CMet be the full subcategory of complete metric spaces. It is well known (see
e.g. [14]) that

(1) for every metric space X there exist a complete metric space X∗ and
a dense embedding γ : X → X∗ such that, for every complete metric
space Y and non-expansive map f : X → Y there exists a (unique)
non-expansive map f∗ : X∗ → Y for which γ ◦f∗ = f;

(2) if f : X → Z is a dense embedding into a complete metric space Z
then Z coincides, up to isometries, with X∗.

Since in Met dense non-expansive maps are epimorphisms, (1) says that CMet
is epi-embedding reflective in Met. It is also well known that the category of
compact Hausdorff spaces is embedding-epireflective in the category of Ty-
chonov spaces, being here X∗ the Stone-Čech compactification of X. However
no property (2) is fulfilled by the latter construction. In this sense we can say

∗Dedicated to Professors Miroslav Hušek and Gerhard Preuss on their sixtieth birthday



144 E. Giuli

that property (2) distinguishes completions from compactifications (see [6] and
[5] for a general treatment of categorical completions).

It is also known (see e.g. [6]) that the category Sob0 of sober T0 spaces
(every irreducible closed set is the closure of a unique point) is epi-embedding
reflective in the category Top0 of all T0 spaces and that property (2) is fulfilled
(i.e., Sob0 is firmly epireflective in Top0 in the terminology of [6]).

On the other hand no non trivial epireflective subcategory of Top consisting
of Hausdorff spaces admits a firm epireflection (cf. [6] Example 1.8(2)).

The aim of this paper is to give sufficient conditions for a class of T0 objects
of a large enough topological category, SSet, whose objects are called affine
sets, to admit a firm epireflection or, as we shall say, to admit completions.

For that we introduce and we study in Section 2 the category SSet of affine
sets and affine maps which properly contains the category Top of topological
spaces and continuous maps.

In Section 3, we analyze the extension from Top to SSet of the ordinary
closure, the b-closure and the Zariski closure (the last two do not coincide in
SSet, while they are the same in Top).

In Section 4 we restrict our considerations to affine sets satisfying a mild
condition (the corresponding full subcategory is denoted by SSET) and there
we introduce and study the so called separated ( = T0) affine sets.

Section 5 contains the main result: a class of separated affine sets admits
completions whenever it is the class of all separated affine sets of a subcategory
of SSET which is stable under affine subsets, disjoint unions and quotients
(i.e., of a hereditary coreflective subcategory of SSET).

Two methods to produce hereditary coreflective subcategories of SSET and
of Top are considered. The first goes back to a paper of Diers [13] (from
which some terminology is derived, e.g., affine set, Zariski closure, etc.) and
the second goes back to a paper of Hušek and the author [16].

Finally two proper classes of examples of classes of T0 spaces admitting
completions are provided:

(a) For each infinite regular cardinal α, the class of all T0 spaces for which
every point in the closure of a subset is also in the closure of a smaller
subset of cardinality less than α. If α is a successor cardinal we obtain
the T0 spaces of tightness less than α.

(b) For each infinite cardinal α, those T0 spaces for which the intersection
of less than α open sets is open.

For categorical terminology see [1] and [20]. For General Topology we refer to
[14].

All the subcategories considered in the paper are full and isomorphism
closed. These are frequently identified with classes of objects defined by a
given property.



On classes of T0 spaces 145

2. The category SSet of affine sets

An affine set over the two point set S = {0, 1} is a pair (X,U) , where X is
a set and U is a subset of the power set P(X). An affine map from (X,U) to
(Y,V) is a function f : X → Y such that f−1(V ) ∈ U for every V ∈ V. SSet
will denote the category of affine sets (over S) and affine maps. The functional
isomorphic description of SSet is as follows: objects are pairs (X,A) where X
is a set and A is a subset of the power set SX and the morphisms from (X,A)
to (Y,B) are functions f : X → Y such that β ◦f ∈A whenever β ∈B.

Both descriptions of SSet will be utilized throughout the paper. We will
denote by F : SSet → Set the obvious forgetful functor.

Proposition 2.1. (SSet,F) is a topological category.

Proof. To show that every F-structured source admits a unique initial lift, let
X be a set, {(Yi,Ui) : i ∈ I} a family of affine sets and {fi : X → Yi : i ∈ I}
a family of functions. Then the subset U = {(fi)−1(V ) : V ∈Ui, i ∈ I} is the
unique initial structure in X for the given data. �

Thus an affine map f : (X,U) → (Y,V) is initial (with respect to F) if
and only if every U ∈ U is of the form f−1V , V ∈ V. As usual an initial
monomorphism will be called an embedding. It is clear that every subset M
of the underlying set X of an affine set (X,U) carries as initial structure the
family V = {U ∩ M : U ∈ U}. In this case we say that (M,V) is an affine
subset of (X,U).

Some consequences (cf. [1, 17] ) of Proposition 2.1 are collected in the

Corollary 2.2.
(i) Every F -structured sink {gi : F(Xi,Ui) → Y, i ∈ I} admits a unique

final lift (i.e., there is a largest affine structure in Y for which all the
gi are affine maps );

(ii) In SSet the epimorphisms are the surjective affine maps and the mono-
morphisms are the injective affine maps;

(iii) In SSet the embeddings coincide with the regular monomorphisms ( =
equalizers of two affine maps );

(iv) Every affine map f admits an essentially unique (surjective, embed-
ding )-factorization. That is: f = mf◦ef for some surjective affine map
ef and embedding mf and, for every commutative square f ◦e = m◦g
with e surjective and m embedding, there exists a (unique) affine map
d such that d◦e = g and m◦d = f;

(v) SSet is complete. Every limit is obtained as an initial lift of the cor-
responding limit in Set.

In particular, the product of a family {(Xi,Ui) : i ∈ I)} of affine sets
is the cartesian product X of the family {(Xi) : i ∈ I} endowed with
the affine structure U = {π−1i (U) : U ∈ Ui, i ∈ I} where πi : X → Xi
are the projections;



146 E. Giuli

(vi) SSet is co-complete. Every colimit is obtained as a final lift of the
corresponding colimit in Set.

Example 2.3. (a) Every closure (hence every topological) space (cf. [11, 9]),
is an affine set and a function between closure (resp. topological) spaces is
continuous if and only if it is affine. Thus both the categories Top of topological
spaces and Cl of closure spaces are fully embedded in SSet;

(b) Affine spaces coincide with so called normal (Boolean) Chu spaces re-
cently introduced by William Pratt as a generalization of Nielsen, Plotkin and
Winskel’s notion of event structure for modelling concurrent computation [19].
Moreover continuous maps between normal (Boolean) Chu spaces coincide with
the affine maps. Thus SSet is a full subcategory of the category ChuS of
(Boolean) Chu spaces and continuous maps.

(c) Following L. M. Brown and M. Diker [3] a texture space is a pair (X,U)
where U is a subset of the power set P(X) which is a complete, completely
distributive lattice with respect to the inclusion, which contains X and ∅,
separates the points of X, and for which meet coincides with intersection and
finite join with union. Thus texture spaces are particular affine spaces. In our
context the natural morphisms between texture spaces are the affine maps. For
suitable morphisms in the class of texture spaces see [4].

Remark 2.4. (1) Following the same lines as the functional description of
SSet, in [15] the category ASet, for every set A, was considered. The above
category coincides with the full subcategory of the category ChuA of Chu
spaces with respect to the set A, consisting of normal Chu spaces [19] and,
for particular instances of A it contains the category Fuz of fuzzy topological
spaces (A = [0, 1]) and the category AP of approach spaces (A = [0,∞]) [15].
The categories of the form ASet fulfil both the proposition and the corollary
above.

(2) The category SSet is not well-fibred even though every set admits a set
of affine structures. In fact the empty set admits two affine structures and
every one-point set admits four structures. This defect can be removed by
assuming that every affine structure contains the empty and the whole set. We
shall consider this full subcategory of SSet, denoted by SSET, in Section 4.

3. Kuratowski and Zariski closure

In this section we extend from Top to SSet the usual (Kuratowski) closure
k, the b-closure in the form introduced by Baron [2] (to characterize the epi-
morphisms in the category Top0 of T0-spaces), and the b-closure in the form
considered by Skula [21]. The latter, called Zariski closure and denoted by z,
in contrast with the situation in Top, does not coincide with Baron closure in
SSet.

We recall that a closure operator c of a topological category (over Set) X is
an assignment, to each subset M of (the underlying set of) any object X of X,
of a subset cXM of X such that



On classes of T0 spaces 147

(c1) M ⊂ cXM;
(c2) cXM ⊂ cXN whenever M ⊂ N;
(c3) c-continuity. For every f : X → Y in X and M subset of X, f(cXM) ⊂

cY (fM).

Note that by property (c2)
(c4) cXM ∪ cXN ⊂ cX(M ∪N).

We will drop X in cXM when no confusion is possible.

The closure operator c is called

(c5) idempotent if cX(cXM) = cXM;
(c6) grounded if cX∅ = ∅;
(c7) additive if cX(M ∪N) = cXM ∪ cXN.
(c8) hereditary if, for every M ⊂ Y ⊂ X, cY M = (cXM) ∩ Y, where Y is

endowed with the initial structure induced by the inclusion Y ⊂ X.
A subset M ⊂ X is called c-closed (respectively c-dense) in X if cXM = M
(respectively cXM = X). A morphism f : X → Y is called c-dense if f(X) is
c-dense in Y and it is called c-closed if it sends c-closed subsets into c-closed
subsets.

An object X is called

(1) c-separated if the diagonal ∆X = {(x,x) : x ∈ X} is c-closed in the
square X2;

(2) absolutely c-closed if it is c-separated and it is c-closed in every c-
separated object in which it can be embedded;

(3) c-compact if, for every object Y , the projection p : X × Y → Y is
c-closed;

(4) c-connected if the diagonal ∆X is c-dense in X2.

It is well known (e.g. see [14]) that in Top, if c is the ordinary closure, then c-
separated means Hausdorff, absolutely c-closed means H-closed and c-compact
means compact in the usual sense (no Hausdorff condition is included). The
c-connected spaces coincide with the irreducible spaces, that is: any disjoint
open sets U,V must satisfy U = ∅ or V = ∅ (cf. [8]).

For the general theory of closure operators we refer to [12, 10] and [7].

If (X,U) is an affine set and x ∈ X we will denote by Ux the family of all
U ∈U such that x ∈ U.

Let (X,U) be an affine set and let M ⊂ X:
(a) the Kuratowski closure of M in (X,U) is defined by

k(X,U)M = {x ∈ X : (∀U ∈Ux)(∃m ∈ M)(m ∈ U)}

(b) the Baron closure of M in (X,U) (cf. [2]) is defined by

b(X,U)M = {x ∈ X : x ∈ k(X,U)(M ∩k(X,U){x})}
= {x ∈ X : (∀U ∈Ux)(∃m ∈ (U ∩M))(∀V ∈Um)(x ∈ V )};



148 E. Giuli

(c) the Zariski closure of M in (X,U) (cf. [21]) is defined by

z(X,U)M = {x ∈ X : (@ U,V ∈U)(U ∩M = V ∩M,x ∈ (U r V )}.

Proposition 3.1.
(i) Always b(X,U)M ⊂ z(X,U)M and b(X,U)M ⊂ k(X,U)M.
(ii) If ∅ ∈U then z(X,U)M ⊂ k(X,U)M.
(iii) If U is a topology then z(X,U)M ⊂ b(X,U)M.

Proof. (i). If x is not in z(X,U)M and U,V ∈U are such that U ∩M = V ∩M
and x ∈ (U r V ), then U ∈Ux and for every m ∈ U ∩M, by m ∈ V , we have
V ∈Um while x is not in V . Consequently x does not belong to b(X,U)M.

The inclusion b(X,U)M ⊂ k(X,U)M is obvious.
(ii). Assume ∅ ∈ U and x not in k(X,U)M. Then there exists U ∈ Ux such

that U ∩M = ∅, ∅ = V ∈U, U ∩M = V ∩M and x ∈ (U rV ); consequently
x is not in z(X,U)M.

(iii). Assume x not in b(X,U)M. Then there exists U ∈ Ux such that, for
each m ∈ U ∩ M there is a Vm ∈ Um with x not in Vm. If U is a topology
then the set V =

⋃
{Vm ∩U : m ∈ (M ∩U)} belongs to U and both conditions

U ∩ M = V ∩ M and x ∈ (U r V ) are fulfilled; consequently x is not in
z(X,U)M. �

Theorem 3.2.
(i) The Kuratowski, Baron and Zariski closure are idempotent and hered-

itary closure operators of SSet.
(ii) The closure operators k and b are grounded in (X,U) if and only if U

is a cover of X. If, in addition, the empty set belongs to U, then z is
grounded.

(iii) The closure operators k, b and z are additive in (X,U) whenever U is
stable under intersection of pairs.

Proof. (i). Properties (c1) and (c2) are trivial and k-continuity of every affine
map (i.e., property (c3)) directly follows from the defining property of affine
map.

Property (c3) for b: if x ∈ b(X,U)M then x ∈ k(X,U)(M ∩ k(X,U){x})
so that fx ∈ k(Y,V)(f(M ∩ k(X,U){x})). Now, by k-continuity of f applied
twice, k(Y,V)(f(M ∩k(X,U){x})) ⊂ k(Y,V)(fM ∩f(k(X,U){x})) ⊂ k(Y,V)(fM ∩
k(Y,V){fx}) consequently fx ∈ b(Y,V)(fM).

Property (c3) for z: assume that y is not z(Y,V)(fM) and let U,V ∈V such
that U ∩ M = V ∩ M and y ∈ (U r V ). Then for every x ∈ f−1y we have
x ∈ f−1U while x is not in f−1V and f−1U ∩ M = f−1V ∩ M which means
that x is not in z(X,U)M, consequently y is not in f(z(X,U)M).

The idempotency of k is clear. Let x ∈ b(X,U)(b(X,U)M) which means x ∈
k(X,U)(b(X,U)M∩k(X,U){x}). Then for every U ∈Ux, U∩b(X,U)M∩k(X,U){x} 6=
∅. Let, for a fixed U, y ∈ U ∩ b(X,U)M ∩k(X,U){x}. For every V ∈Uy, by y ∈
b(X,U)M, V ∩M∩k(X,U){y} 6= ∅ and by y ∈ k(X,U){x}, k(X,U){y}⊂ k(X,U){x}



On classes of T0 spaces 149

consequently V ∩M ∩k(X,U){x} 6= ∅. Now U ∈Uy consequently x ∈ b(X,U)M.
This shows that b is idempotent. For the idempotency of z note that if U,V ∈U
satisfy U ∩ M = V ∩ M then also U ∩ z(X,U)M = V ∩ z(X,U)M, so that if x
is not in z(X,U)M then there exist U,V ∈ U such that (U ∩ M = V ∩ M,
hence) U ∩z(X,U)M = V ∩z(X,U)M and x ∈ (U rV ), consequently x is not in
z(X,U)(z(X,U)M).

The hereditariness directly follows from the fact that the affine structure in
an affine subset Y of an affine set (X,U) is {U ∩Y : U ∈U}.

(ii). k(X,U)∅ = ∅ if and only if U is a cover is clear. Then the remaining
part of (ii) follows from k(X,U)∅ = b(X,U)∅ ⊂ z(X,U)∅.

(iii). This simple example shows that b, z and k are not additive. Let X =
{1, 2, 3} and let U = {{0, 1},{0, 2},∅}. For each M ⊂ X, z(X,U)M ⊂ k(X,U)M,
by ∅ ∈U, b(X,U){i} = k(X,U){i}, for i = 1, 2, and b(X,U){1, 2} = k(X,U){1, 2} =
X. Hence z(X,U){1, 2} = X.

For additivity, one inclusion is (c4), and the other directly follows from the
stability under intersection of pairs of U. �

The next result follows from the idempotency and hereditariness of our closure
operators (see [10, 12]).

Corollary 3.3. Every affine map admits an essentially unique (c-dense, c-
closed embedding ) factorization for c = k,b,z.

Proof. For a given f : X → Y denote by ef : X → cY (fX) the codomain
restriction of f to cY (fX) and by mf : cY (fX) → Y the inclusion map. By
c idempotent, cY (fX) is c-closed and, by c hereditary, ef is c-dense. Thus
mf ◦ef is a (c-dense map, c-closed embedding)-factorization of f.

Let now f ◦e = m◦g (e : X → Y,m : Z → T) be a commutative square in
SSet, with e a c-dense affine map and m a c-closed embedding, and let y ∈ Y .
Since y is in the c-closure of eX in Y by assumption, then, by c-continuity of
f, fy is in the c-closure of f(eX) = (f ◦e)X. By commutativity, fy is then in
the c-closure of (m(gX), hence) mZ. Now mZ is c-closed in T by assumption,
so that fy ∈ mZ, consequently, by injectivity of m, there is unique z ∈ Z such
that mz = fy. Set dy = z. Then d is an affine map since m ◦ d = f and m
is initial by assumption. Moreover d is the unique affine map satisfying both
m◦d = f and d◦e = g since m is injective by assumption. �

Remark 3.4. It should be noted that in the previous proof we have not used
the full power of hereditariness of our closures. Indeed what we need is that
every subset is c-dense in its closure. That property, called weak hereditariness,
is, in the presence of idempotency, weaker than hereditariness (see e.g., [12]).

4. Separated (= T0) affine sets

In this section we will refer to the functional description of SSet



150 E. Giuli

The extension of the ordinary closure k of Top to SSet has no interest for
the development of such basic topological notions as separation, compactness,
absolute closedness and connectedness. Indeed, by the form of products in
SSet as explained in Section 2, an affine set is k-separated if and only if it
is absolutely closed if and only if it has at most one point. Moreover every
affine set is k-compact since every projection is k-closed and every affine set is
k-connected since the diagonal of every affine set is k-dense in the square of
the affine set.

The aim of the last two sections is to show that the above topological notions
are not trivial in SSet if we refer to the Zariski (or Baron) closure.

For that we restrict our considerations to those affine sets whose affine struc-
ture contains the two constant functions 0 and 1. The corresponding full
subcategory, denoted by SSET, which is topological too, has many pleasant
properties not shared by SSet. Among others,

(1) it is well-fibred, i.e. every constant function is affine or, equivalently,
the affine sets with at most one point have unique affine structure;

(2) our closures are grounded there;
(3) a nonempty affine set is indiscrete if and only if its structure consists

of constant functions.
In what follows S will denote the two-point set {0, 1} endowed with the affine
structure A = {0, 1, idS} (U = {∅,S,{1}} in the subset description). S will
be called the Sierpinski affine set.

An affine set (X,A) is called separated if A separates the points of X.
SSET0 will denote the full subcategory of separated affine sets.
Clearly the Sierpinski affine set S is separated.
The following result has a trivial proof but plays an important role:

Lemma 4.1. A function f : X → S is an affine map between (X,A) and S if
and only if f ∈A.

Proposition 4.2. For an affine set (X,A) these are equivalent:
(i) (X,A) is separated;
(ii) every affine map f : I2 → (X,A) is constant, where I2 is a two-point

indiscrete affine set;
(iii) ∆X is z-closed in (X,A) × (X,A);
(iv) ∆X is b-closed in (X,A) × (X,A);
(v) (X,A) is an affine subset of a product of copies of the Sierpinski affine

set S.

Proof. (i)⇔(ii). The existence of a non constant affine map from I2 to (X,A)
is equivalent to saying that there are distinct points x and y in X such that
α(x) = α(y) for every α ∈ A, which is equivalent to saying that (X,A) is not
separated.

(i)⇒(iii). If (x,y) ∈ X2r∆X then, by assumption, there is α ∈A such that
α(x) 6= α(y). Then the two affine maps α◦p1,α◦p2 : X ×X → X coincide in



On classes of T0 spaces 151

∆X and do not coincide on (x,y) which says that (x,y) is not in the z-closure
of ∆X. Consequently ∆X is z-closed.

(iii)⇒(iv). It follows from Proposition 3.1 (i).
(iv)⇒(ii). Assume (X,A) is not separated and let x 6= y in X such that,

for each α ∈ A, α(x) = α(y). Taking into account that the affine structure of
X ×X consists of functions of the form α◦pi, i = 1, 2, it is easy to verify that
the point (x,y) is in the b-closure of ∆.

(i)⇒(v). Let SA be the product of A copies of S and let
φ : (X,A) → SA

be the map whose components are the elements of A. By virtue of Lemma 4.1,
φ is affine and initial, and it is (injective, hence) an embedding if (and only if)
(X,A) is separated.

(v)⇒(i). Separation is clearly a productive and hereditary property. �

Remark 4.3. (1) The equivalence (i)⇔(ii) in the above proposition says that
the separated affine sets coincide with the so called T0-objects of the well fibred
topological category SSET, see [18]. In particular SSET0 is the largest epire-
flective non bireflective subcategory of SSET. Moreover (i)⇔(v) says that
SSET is simply cogenerated by the Sierpinski affine set S.

(2) The SSET0-reflection r of an affine set (X,A) is the restriction to the
image of the affine map φ. Indeed let (Y,B) be a separated affine set, let

ψ : (Y,B) → SB

be the canonical map, let f : (X,A) → (Y,B) be any affine map and let f′ :
SA → SB be the affine map associated to f. Then, denoting by k : φ(X) → SA
the inclusion, we obtain the commutative square (f′ ◦ k) ◦ r = ψ ◦ f. Now
r is surjective and ψ is an embedding since (Y,B) is separated, so that, by
Corollary 2.2 (iv) there exists a (unique) affine map d : φ(X) → Y such that
d◦r = f.

(3) Since the SSET0-reflections are initial (see the proof of (i)⇒(v) in
Proposition 4.2) then the category SSET is universal in the sense of Marny
[18].

Corollary 4.4.
(i) In SSET0 the epimorphisms are precisely the z-dense affine maps and

the regular monomorphisms are precisely the z-closed embeddings.
(ii) In SSET0 every affine map admits an essentially unique factorization

by a z-dense (respectively b-dense) affine map followed by a z-closed
(respectively, b-closed ) embedding.

Proof. (i). By Lemma 4.1 the morphisms from an affine set (X,A) to the
Sierpinski affine set are precisely the elements of the structure of A. On the
other hand our Zariski closure of a subset M is obtained by intersecting all the
equalizers, of pairs in A, containing M, so, equivalently intersecting equaliz-
ers of pairs of affine maps into S. Then, z being the regular closure operator



152 E. Giuli

induced by S hence, in virtue of (i)⇔(v) for SSET0, it gives the epimor-
phisms and the regular monomorphisms (for that we use weak hereditariness)
in SSET0 as in (i) (see [10]).

(ii). Every affine subset of a separated affine set is separated, so the state-
ment follows from Corollary 3.3. �

5. Completions

An affine set (X,A) is called z-injective if it is injective with respect to z-
dense embeddings. That is: for every z-dense embedding m : (M,C) → (Y,B)
and affine map f : (M,C) → (X,A) there exists an affine map f′ : (Y,B) →
(X,A) such that f′ ◦m = f.

By a standard argument z-injectivity is preserved by products.

Since embeddings are initial affine maps, by Lemma 4.1 the Sierpinski affine
set is z-injective.

Proposition 5.1. For a separated affine set (X,A) these are equivalent:
(i) (X,A) is z-injective;
(ii) (X,A) is absolutely z-closed;
(iii) The canonical map φ : (X,A) → SA is a z-closed embedding.

Proof. (i)⇒(ii). Let (X,A) be separated and z-injective, let k : (X,A) →
(Y,B) be an embedding with (Y,B) separated and m◦e the (z-dense, z-closed
embedding)-factorization of k (see Corollary 4.4 (ii)). Then, since (X,A) is
z-injective, there is an affine map g from the codomain of e into (X,A) such
that g ◦ e = 1X, which says that e is a section, hence an isomorphism since it
is an epimorphism in SSET0.

(ii)⇒(iii). Trivial.
(iii)⇒(i). Assume that the canonical map φ : (X,A) → SA is a z-closed

embedding, let k : (M,C) → (Y,B) be a z-dense embedding and f : (M,C) →
(X,A) any affine map. Since the elements of C are the restrictions to M of
those of B, there is an affine map f′′ : SB → SA such that (f′′ ◦ψ) ◦k = φ◦f
consequently, by Corollary 4.4 (ii), there is an affine map f′ : (Y,B) → (X,A)
satisfying, in particular, f′ ◦k = f. �

We are now ready to prove that SSET0 admits completions.

Theorem 5.2. The full subcategory CSSET0 of SSET0 consisting of all the
absolutely z-closed affine sets is firmly epireflective in SSET0.

Proof. The CSSET0-reflection s of a separated affine set (X,A) is the restric-
tion to the z-closure in SA of the image of the affine map φ. The proof uses
the same argument as Remark 4.3 (2).

The property (c2) directly follows from the fact that the Sierpinski affine set
is a z-injective cogenerator of SSET0. �



On classes of T0 spaces 153

Recall that a subcategory X of a category Y is called coreflective in Y if for
every object Y of Y there exist an object Ŷ in X and a morphism s : Ŷ → Y
such that, for every X ∈ X and morphism f : X → Y there exists a unique
f′ : X → Ŷ satisfying s◦f′ = f.

If Y is topological over Set (as is our SSET), then the coreflection maps s
are bimorphisms, so we may assume, in our context, that ̂(X,A) is X endowed
with an affine structure finer than or equal to A and that the coreflection maps
are identities. If a coreflective subcategory of SSET is stable under affine
subsets we shall say that it is hereditarily coreflective.

Let X be a hereditary coreflective subcategory of SSET and let us denote
by T0X the subcategory of its separated affine sets. We shall show that T0X
admits completions.

Lemma 5.3. If X is a hereditarily coreflective subcategory of SSET then:
(i) Ŝ is separated and z-injective in X.
(ii) T0X is cogenerated by Ŝ in X.

Proof. (i). Since Ŝ is a finer modification of S it is separated. Let k : X → Y
be a z-dense embedding with X,Y ∈ X and let f : Y → Ŝ any affine map.
Since S is z-injective there exists f′′ : Y → S with f′′ ◦k = s◦f (s : Ŝ → S).
Consequently, by the universal property of coreflections, there is f′ : Y → S
such that s◦f′ = f′′ so that f′ ◦k = f , by injectivity of s.

(ii). It is clear that a function from an affine set in X to S is affine if and only
if it is so into Ŝ. Consequently the canonical map φ : (X,A) → ŜA remains an
embedding whenever (X,A) is in X and the product ŜA is taken in X. �

Theorem 5.4. If X is a hereditary coreflective subcategory of SSET then the
affine sets which are affine subsets of products (taken in X) of copies of Ŝ form
a firm epireflective subcategory of T0X.

Proof. The proof follows the lines of proof of Theorem 5.2 (and Remark 4.3 (2)).
�

There is a general method to produce hereditary coreflective subcategories
of SSET:

Recall that an algebra structure in a set A is a family of functions

Ω = {ωT : AT → A}
where T runs in a given class of sets. Then for every set X, by point-wise
extension, the powerset AX carries an algebra structure. If Ω is an algebra
structure in the two-point set S we denote by SSET(Ω) the subcategory of
SSET consisting of those affine sets (X,A) for which A is a Ω-subalgebra of
the function algebra SX.

It is easy to show that, for every algebra structure Ω in S the correspond-
ing subcategory SSET(Ω) of affine sets over the algebra (S, Ω) is hereditarily
coreflective in SSET.



154 E. Giuli

In this way, for suitable Ω, we obtain among others, the topological categories
CS of closure spaces (cf. [9]), Top of topological spaces and PrOS of pre-
ordered sets.

For CS an internal characterization of the complete ( = absolutely closed) T0
spaces is given in [9]; in Top0 they are the sober T0 spaces (see the introduction)
and in PrOS it is easy to see that they are the partially ordered sets.

We do not know any example of a hereditary coreflective subcategory of
SSET which is not of the form SSET(Ω).

There is a second general method to produce hereditary coreflective sub-
categories, e.g. in Top, which is described in [16]. In that paper two proper
classes of hereditary coreflective subcategories of Top are produced: for every
regular cardinal α,

Alex (α) = {X ∈ Top : intersection of less than α open sets in X is open},
Tight (α) = {X ∈ Top : if x ∈ B̄ ⊂ X then x ∈ Ā for some A ⊂ B, |A| < α}.

While it is easy to show that every category in the first class is of the form
SSET(Ω) we do not know if (at least) one of the categories of the second class
is of the form SSET(Ω).

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On classes of T0 spaces 155

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Received March 2002

Revised March 2003

Eraldo Giuli

Department of Mathematics,
University of L’Aquila,
67100 L’Aquila,
Italy.

E-mail address : giuli@univaq.it


	On classes of T0 spaces admitting completions. By E. Giuli