Applied Categorical Structures 1: 285-295, 1993. © 1993 KluwerAcademic Publishers. Printed in the Netherlands.

285

Hausdorff Separation in Categories M. MANUEL CLEMENTINO* Departamento de Mathem(~tica, Universidade de Coimbra, 3000 Coimbra, Portugal (Received: 3 November 1992; accepted: 24 March 1993)

Abstract. Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion of T2-object. We discuss the properties of Tz-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.

Mathematics Subject Classifications (1991). 18B30, 18A40, 54A05, 54B30, 54D10.

Introduction Several authors have studied separated objects, which have their origin in the following characterization of Hausdorff spaces: a space X is Hausdorff if and only if A x = {(x, x)tx E X} is closed in X × X. These objects depend on a fixed closure operator (see, for instance, [12] and [9]), or even on a factorization system (see [13] and [10]). In this paper we introduce and study another categorical generalization of the Hausdorff separation axiom of topological spaces which resembles more directly the idea of separating points. In order to do that, having a fixed class of subobjects, we introduce a relative notion of point which depends on the choice of a class 79 of objects. This class 79 may be quite general: it just has to be closed under direct images. The consideration of points in a category equipped with a closure operator allows us to define and to study the notion of Tz-object as indicated. Under certain natural conditions T2 and separated objects coincide, but in general there are no reasons to expect this to hold since Tz-objects depend not only on the closure operator but also on the points previously chosen. This, from our point of view, is not a disadvantage of the notion of T2-object. On the contrary, it allows us to substantiate certain phenomena which could not be described by the notion of separatedness. For instance, Willard observes that: " . . . compact subsets o f a topological space behave like p o i n t s . . • "(see [15], Sec. 17). In fact, our notion of Tz-object leads to the same class of spaces (namely: Hausdorff spaces), regardless of whether our points are chosen to be the compact sets or the usual points of a topological space (see Example 2.3.1 below). • The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N~I.C.-Instituto Nacional de Investigag~o Cienfffica.

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In Section 1 we present the categorical setting in which we shall be working. First we describe the basic assumptions on subobjects, and we introduce the concept of weak complement and its general properties. Then we introduce the notion of point and discuss some desirable properties for points and their influence on the behaviour of weak complements. Having fixed the notions of subobject, point and closure operator, in Section 2 we introduce T2-objects and present some relevant examples. Then, making use of the points and the closure operator, we consider a functor Pt from 2( to the category CS of closure spaces. Analysing the behaviour of this functor with respect to T2-objects (Theorem 3.1), we deduce important properties of the subcategory T2 of T2-objects, namely its reflectiveness under mild conditions on 2( (Corollary 3.2). Finally, in Section 4, we give sufficient conditions for the notions of T2 and separated object to coincide. These conditions depend again on the functor Pt (Theorem 4.2 and Corollary 4.3).

1. Subobjects, Complements and Points In a category 2( we consider a fixed class A4 of 2(-morphisms closed under composition and containing all isomorphisms. Furthermore, we assume that 2( is .M-complete (i.e. the pullback of an .M-morphism along any morphism and multiple pullbacks of (arbitrary) families of .M-morphisms exist and belong to

M). These assumptions have several interesting consequences. Namely: (i) A4 is a class of monomorphisms (see, for instance, [14]); (ii) there exists a class C of 2(-morphisms such that (E, h4) is a factorization system (i.e. each 2(-morphism has an (E, .M)-factorization and the (E, .M)diagonalization property holds - cf. [14]); (iii) for every 2(-object X, the class A4/x of .M-morphisms with codomain X equipped with the preorder < defined by m <_ r~ :¢~ 3k : n • k = m is complete; (iv) each 2(-morphism f : X --+ Y defines functors f - l ( _ ) . .A4/y --+ .A4/x and f ( - ) : .A4/x -~ A4/y given by pullback and (E, .A4)-factorization, respectively, f ( - ) being left adjoint to f - 1 (_) (considering .M/x and .A4/y as comma categories). In A 4 / x we can consider the equivalence relation -~ given by m -~ n if and only if m <_ ~z and n <_ m. Then the class . M / x modulo ~, which we denote by Sub(X), is partially ordered. We shall use the terminology and notations of lattice theory for both A4/x and Sub(X). The elements of Sub(X) (or just their representatives in A 4 / x ) will be called subobjects of X.

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From the properties of A d / x already mentioned it follows that S u b ( X ) has a least element, which we call the trivial subobject of X and denote by 0 x : 0 x --e X. Subobjects are not in general complemented. Nevertheless, the notion of weak complement that we introduce below will be useful in the sequel. DEFINITION 1.1. For each subobject m of X, the subobject

V {"CM/xlrA

-Ox}

will be called the weak complement of m and denoted by m °. It is easily seen that

• (0x) ° ~= 1x; ( 1X ) 0 ~ = Ox; • (Vm, n ~ M / x ) m < n =~ m ° >_ no; • (Vm ~ M I x ) ( m v m°) ° -~ Ox. Furthermore, for m , n C A A / x and r E AA/M, if r A m - l ( n ) ~ 0M then m . r A n ~- Ox, hence m . r ~- re(r) < n °. Then r <_ m - l ( m • r) <_ m - l ( n ° ) , and we conclude that • (V~'/,,/Z e

,All/X)

(~7~-l(n)) 0 (___m - l ( n 0 ) .

Let us introduce now our main tool: the definition of points for each object X of X. We consider a fixed class 79 of X-objects closed under g-morphisms (i.e. if e : P ~ Q is an g-morphism and P belongs to 79, then Q also belongs to 79). We shall denote by 79x the class of Ad/x-morphisms with domain in 79. DEFINITION 1.2. For each object X the elements of S u b ( X ) with domain in 7:' will be called points of X. The class of points of X will be denoted by p t ( X ) . We remark that, from the closedness under g-morphisms of 79, it follows that, for every X-morphism f • X ~ Y and every point z of X, the subobject f ( z ) is a point of Y. In topological spaces, the usual points are 'small' and exist 'everywhere', in the following sense: (S)

( V X c X ) ( V m ~ M / x ) ( V z ~ P x ) m <_ ~c ~ m ~- Ox or m ~ z;

(E)

( vx e

c

M/x)

r'x t • _<

These conditions are different in nature: the former controls the size of points and the latter their existence. Conditions (S) and (E) will be very useful in the sequel. We first study their impact on weak complements.

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PROPOSITION 1.3. Condition (E) is equivalent to: (VX e X ) ( V m • M / x ) m °~- V { x e 7 9 x I x A m ~ - - O x } .

Proof. I f n ~ V{x • 79x ] x A m -------0x}, then n < m °. Moreover, under(E), if r A m ~ 0 x then r < n since, for each y • 79x such that y < r, we have that y < n. Therefore, if(E) holds, then m ° --- V{x • 79x I x A m TM 0x}. Conversely, assume that m ° ---- V{x • 79x ] x A m TM 0 x } for each subobject m of each X-object X. Then, l x - (0x) ° ~ V{xlx E 79x} for each X, and it is easily checked that this condition is equivalent to (E). [] In adding condition (S) one derives further interesting properties. PROPOSITION 1.4. Under conditions (E) and (S), the following assertions hold: (a) (VX • X)(Vm • A 4 / x ) m V m ° ~- l x ; (b) ( V f " X --+ Y • M o r X ) ( V n • J k 4 1 y ) ( f - l ( n ) ) ° < f - l ( n ° ) .

Proof. If x is a point of X, then from (S) one obtains x < m or x A m ~ 0 x , hence, x < m or x <_ m °. Therefore, from (E) it follows that m V m ° = I x , that is, (a) holds. In order to prove (b) one has to verify only that, for each x E 79x, if x A f - l ( n ) -~ 0 x , then f ( x ) A n ~- Or. Let x be a point of X. From (S) it follows that f ( x ) A n ~- Or" or f ( x ) A n ~- f ( x ) , in which case we have that x < f - l ( n ) . Hence, if x A f - l ( n ) ~- Ox, then f ( x ) A n ~- Oy or x = x A f - l ( n ) -------0x. Therefore, in both situations we have f ( x ) A n ~ Or as claimed. [] Finally we point out that, i f p t ( X ) is a set for each X-object X , then the assignment X H p t ( X ) defines a functor pt : X --+ Set which, for a morphism f : X ---+ Y gives the map

p t ( f ) : pt(x)

x

pt(Y), f(x).

2. T2-Objects From now on we shall be working in the context described in the previous section: we consider an M-complete category X, .A4 being a class of X-morphisms closed under composition and containing the isomorphisms, and a fixed class of X-objects 79 closed under E-morphisms. Moreover, we consider a closure operatorC on X with respect to A4 in the sense of [5], that is: a family C = (cx : A 4 / x --+ A 4 / x ) x e x of extensive and monotone maps e s such that f ( c x ( m ) ) < c y ( f ( m ) ) for each X-morphism f : X ---+ Y a n d subobject m of X. More on closure operators can be found on [5] and [6].

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In this section we shall be concerned with the notion of Tz-object. DEFINITION 2.1. An X-object X is said to be a T2-object if

(vz, y

e ~'x)z

A

y ~ Ox => (~m ~ 3 4 / x )

•

z

A

c x ( m ) -~ y

A

c x ( m °) -~ Ox.

We shall denote by T2 the (full) subcategory of T2-objects of X and by (T2) the above condition. PROPOSITION 2.2. The subcategory 7-2 is closed under subobjects. Proof Let Y be a T2-object, m : X ~ Y a subobject of Y and x and y points of X such that x/X y -- 0 s . We thus have re(x) A re(y) -- m . x A m . y ~- 0y and then, by (T2), there exists n E 3 4 / y such that m . x A cy(n) ~ m . y A cy(n °) ~- Oy. Now, from the properties of m -1 ( - ) , C and 0 °, we deduce that

x A cx(~-l(n))

_<_.~ -l(.~ . x) A ~ - l ( c v ( n ) ) - ~ - l ( 0 y ) -~ ox

and

yAcx((m-l(n)) O) <

y A ~x(~-l(~°)) _< . ~ - ~ ( ~ . y)/x ~ - l ( ~ y ( ~ o ) ) ___~ - l ( o r ) u Ox.

Therefore, X belongs to T2 as claimed.

[]

EXAMPLES 2.3. 1. In the category Top of topological spaces and continuous maps, let 34 be the class of embeddings and C the usual (Kuratowski) closure operator. (a) Consider 79 = { X t X is a singleton space}. In this situation it is clear that conditions (E) and (S) hold since points of a space correspond bijectively to the elements of its underlying set. Therefore T2-objects coincide with Hausdorff

spaces. (b) Consider 7) = {X IX is a compact space}. For this choice points fulfill (E) but not (S). Also in this case we have that T2 is the category of Hausdorffspaces (see, for instance, [151, 17.6 and 17.B). 2. Consider the category 7)rTop of Cech (= pretopological) spaces. We recall that objects in P r T o p are sets equipped with a grounded and additive closure operation (i.e. pairs (X, c : I?X ~ I?X) where X is a set, IPX is its power set, and c is an extensive map such that c(O) = ~ and c(M U N) = c(M) U c(N) for each M and N in I?X), and morphisms are continuous maps with respect to the closure operations (i.e. a map f : (X, c) ~ (Y, d) is a morphisms in T'rTop if f(c(M)) C_ d ( f ( M ) ) for each subset M of X). Let AA be the class of extremal monomorphisms and C the natural closure operator in 79rTop, that is, c(x,c ) = c for each pretopological space (X, c).

290

M. M A N U E L C L E M E N T I N O

Consider 79 = {(X, c) I X is a singleton}. It is obvious that points of (X, c) correspond bijectively to elements of X. Hence conditions (E) and (S) are obviously fulfilled. In this situation T2-objects coincide with separated closure spaces in the sense of (~ech [1] (cf. 27A.1). 3. In the category Top. of pointed topological spaces and point-preserving continuous maps, let AA be the class of extremal monomorphisms (hence Sub(X, xo) describes the subsets of X containing x0) and C = (C(x,xo))(X,zo)ET-op* be the closure operator induced by the Kuratowski closure operator, i.e. C(X,xo)(M) = 217/ for each subset M of X containing x0. Let 79 = {(X, x0) [ card X < 2}. Hence points of a pointed topological space (X, x0) correspond bijectively to the subsets of X containing x0 and having at most two elements. It is easily checked that they fulfill (E) and (S). Then T2 = { (X, xo) I {xo } = {xo } and X \ {xo } is Hausdorff}. 4. In the category Top/B of fibrewise topological spaces over B and continuous fibrewise maps, let A/[ be the class of extremal monomorphisms (hence Sub(X) may be identified with the power set of X), and C = (¢(X,p))(X,p)ETop/B be the closure operator defined by: (V(X,p) E Top/B)(VM C_X)c(x,p)(M ) = 2kf Mp-l(p(M)), with hT/denoting the usual closure of M in X. Consider the points defined by 79 = {(X,p)IX is a singleton space}. As in 1 (a), the set pt(X, p) can be identified with the underlying set of X, and conditions (E) and (S) are clearly fulfilled. In this situation T2-objects coincide withfibrewise Hausdorffspaces in the sense of James [11]. 5. Let A' be the category Top a of G-spaces (see [3]), .AA the class of extremal monomorphisms (we thus have that Sub ((X, H)) is the set of H-invariant subsets of X), and C the closure operator defined by

(V(X, II) E TopG)(VM E Sub(X,H))

c(M) = M.

Consider the points defined by the class 79 of epimorphic images of (G, "/. Then pt((X, YI)) coincides with the set X/cn of orbits of X, and it is easily verified that (E) and (S) hold. In this case we have that IX, H) is a T2-object if and only if Cn is a closed subset of X × X (see [3], Proposition 1.3.10). 6. Let X' be the category AbGrp of abelian groups, .M the class of monomorphisms (hence Sub(X) may be identifed with the set of subgroups of X) and C the closure operator which, for each subgroup of an abelian group X, gives cx(M) := {x E X [ 3n E 51 : nx E M}. (We point out that C is the regular

291

HAUSDORFF SEPARATION IN CATEGORIES

closure operator induced by the (full) subcategory of torsion-free abelian groups; for general information on regular closure operators see, for instance, [6].) Let us choose for T' the class of cyclic groups. In this case points satisfy (E) but not (S). Concerning T2-objects one derives that

7"2 = { X I X is torsion-free and Z × Z is not a subgroup of X } , that is, an abelian group is 7"2 if and only if it is torsion-free and its rank (in the sense of [7]) is less or equal to 1. 7. Consider the category Grph of directed graphs, whose objects are pairs (X, A), where X is a set and A a subset of X 2, and having as morphisms from (X, A) to (Y, B) maps f : X ~ Y such that (f(x), f(x')) E B whenever (x, x') E A. Let A/[ be the class of extremal monomorphisms of ~rph (note that Sub(X, A) may be identified with the power set of X), and C the usual closure operator, that is, for each object (X, A) and each subset M of X, C(X,A)(M) := {x E X [ x E Mor(ByEM:(y,x) CA)}. L e t P : { ( X , A ) [ c a r d X : 1} ( : {(X,A) I(X,A) "~ ({.},(~)or (X,A) ( { * }, { ( *, *) } ) ). Hence points of a directed graph ( X, A ) correspond bijectively to elements of X, and it is easily seen that they fulfill (E) and (S). Then

7"2 = {(X,A) l (x.~y) e A =~ x : y}. 3. T h e F u n c t o r P t a n d T2-Objects

The main goal of this section is the study of properties of the subcategory T2 making use of a lifting Pt of the functor pt to the category CS of closure spaces. We recall that CS is the category of sets equipped with closure operations and continuous maps with respect to these closure operations (notice that the category PrTop studied in Example 2.3.2 is a full subcategory of CS). From now on we assume that pt(X) is a set for each ?f-object X. Next we shall consider, for each ?f-object X, a closure operation on pt(X) - which depends on the closure operator C - in such a way that the maps pt(f) (for every f 6 Mor?f) remain 'continuous morphisms'. For that we consider the monotone maps

Cx : P(pt(X)) --~ Sub(X) A ~

C x ( A ) : = V{xlx 6 A}

and

Cx : Sub(X) {xlx <_ with P(pt(X)) denoting the powerset of pt(X). It is easily seen that Cx • ~Px _< ls~b(x) and ¢ x • Cx >_ l~(pt(X)), that is, as functors, Cx is left adjoint to Cx.

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M. MANUEL CLEMENTINO

Moreover, the equality Cx • C x = lsub(X) holds for every object X if and only if the points verify (E). Now, for each X-object X, consider the map

: e(pt(x))

-+ e ( p t ( x ) ) ,

A H ")'x(A)= Cx(cx(¢x(A))) which is easily seen to be monotone and extensive, that is, a closure operation. L e t f : X ---+ Y be an X-morphism. For each m E Sub(X) a n d A E lP(pt(X)), one can prove that

f ( ¢ x ( A ) ) < Cy(pt(f)(A)) and

pt(f )(~bx(m) ) = C y ( f (m) ) hold. From these conditions it is easy to deduce thatpt(f) (Vx (A)) C__~v (pt(f) (A)) for every subset A ofpt(X). Hence, the map pt(f) is a morphism in CS. In conclusion, the correspondences above define a functor

Pt : X --+ CS X ~-~ P t ( X ) = (pt(X),Tx) f ~-+ P t ( f ) = pt(f) which clearly fulfils the equality I • l" Pt = pt, where I " I denotes the forgetful functor CS --+ Set. Whenever the points verify (E) and (S), the functor Pt is a relevant tool for the study of T2-objects. THEOREM 3.1. Under conditions (E) and (S) the functor Pt preserves and reflects

Tz-objects. Proof First we remark that, under (E) and (S), the relations ( ¢ x ( A ) ) ° < C x ( A °) and (¢x(m)) ° C_ C x ( m °) hold for every X E X, m E A 4 / x and A E l?(pt(X)). Now, let X be a Tz-object of X. If x, y E pt(X) and x ~ y, then from (E) and (S) it follows that x and y are non-trivial subobjects and that x A y ------0x. Hence there exists a subobject m of X such that x A cx (m) ~- y A cx (m °) ~- Ox, by (T2). For A -- C x ( m ) we have that ~/x(A) c zbx(cx(m)) by definition of'yx and then x • "yx(A), and ~/x(A °) C_ Vx(Ox(m°)) C_ Cx(cx(m°)), hence y ~[ vx(A°). Therefore, P t ( X ) is a T2-object. Conversely, if Y is an X-object such that Pt(Y) is T2 and the points x and y of Y satisfy x A y ~ 0y, then, either x -------y ~ 0y, in which case (T2) is trivially verified, or x ~ y. In the latter case there exists a subset B of pt(Y) such that x ~ 7y(B) and y ~ 7y(B°), by hypothesis. Therefore, for n = C y ( B ) , one can

HAUSDORFF SEPARATION IN CATEGORIES

easily verify that x A cy, (n) ~ y A cy (n °) ~- 0y, that is, Y is a T2-object.

293 []

COROLLARY 3.2. Assume that the conditions (S) and (E) hold. If the functor P t preserves products (monomorphisms, monosources resp.), then the subcategory 7"2 is closed under products (monomorphisms, monosources resp.). Proof Since it is clear that the subcategory of T2-objects of CS is closed under monosources, the required result follows directly from Theorem 3.1. [] We remark that this result has an interesting consequence: under mild conditions on 2,, the subcategory T2 is reflective whenever the functor P t preserves monosources. For instance, if 2, admits the factorization system (extremal epi, monosource) for sources, then from the preservation of monosources by P t it follows that Tz is an extremally epireflective subcategory of 2".

4. Relationships between T2 and Separated Objects Let 2' be a category equipped with a closure operator C with respect to a class A4 containing the regular monomorphisms of 2". An 2,-object is called (C)-separated whenever the diagonal A x : X --+ X x X is (C)-closed. We shall denote by Sep the (full) subcategory of separated objects of 2". Having defined points in the category 2", a seemingly natural question in this study is whether the functor P t : 2( --+ CS preserves and reflects separated objects. However a quick examination reveals that in the category of closure spaces an object (X, d) is separated with respect to the natural closure operator iff the set X has at most one element. This strange fact is due to the behaviour of products in CS: the closure operation of the product ((X, c), (Po~)A) in CS of a family (X,~, Co~)o~EAof closure spaces is defined as follows

(vm c X)c(M) = II c.(p.(M)). e~EA We therefore wish to restrict the codomain of the functor P t to the full subcategory 79rTop where products behave more naturally. PROPOSITION 4.1. Let 2", A4, C and 79 be as in Section 2. If (a) the closure operator C is additive; (b) for every X in 2(, ~bx(cx(Ox)) = O; (c) for every X in 2(, the restriction of ~bx to closures of subobjects preserves finite joins, thatis, (Vx E 79x) (Vm, n E A 4 / x ) x <_ c x ( m ) V cx(n) =~ x _< c x ( m ) o r x <_ cx(n),

then P t ( X ) is a pretopological space for each X in 2". Conditions (a)-(c) are in fact necessary for this result provided (E) holds.

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M. MANUEL CLEMENTINO

Proof First we remark that the closure space P t ( X ) is a pretopological space iff it verifies the two conditions below: • 7 x ( O ) = O;

• (VA, B C_pt(X))~/x(A U B) = ~/x(A) U "yx(B). Let us first prove that P t ( X ) is a pretopological space whenever conditions (a)-(c) hold. From (a) it follows that "Tx (O) = (~. That "yx(A U B) = "yx(A) U 7 x ( B ) for each A and B in ?(pt(X)) follows from (b) and (c) and the definition of 7x. Now let us check that, under (E), conditions (a)-(c) are necessary. Condition (a) is immediate since 0 = "7x(0) = ~bx(cx(qSx((~))). To prove the remaining conditions one has to verify only that, under condition (E), if P t ( X ) is a pretopological space for each X-object X, then the equalities V

x(cx(m v

=

v

= V ; x ( c x ( m ) ) u Cx(cx(

))

hold for every object X and subobjects m and n of X.

[]

From now on we assume that X, Ad, C and 79 are such that the image of Pt lives in 79rTop. THEOREM 4.2. If (E) is verified and if the functor Pt : X ~ 79rTop preserves finite products, then P t preserves and reflects separated objects. Proof Let X be an X-object. Since Pt preserves finite products, the diagonal A p t ( x ) : P t ( X ) ~ P t ( X ) × P t ( X ) is isomorphic to P t ( A x ) , where A x is the diagonal of X. We thus have "~x (Apt(x)) = ~bx (cx ( A x ) ) , hence

~ X ( A P t ( X ) ) ---- A P t ( X ) "~ ',. ~ X ( C X ( A X ) ) ~ -:

',- c x ( A x )

= ~/)X(AX)

= Ax,

since Cx is injective by condition (E). We thus have that P t ( X ) is separated iff X is separated.

[]

Observing that in P r T o p T2 and separated objects coincide (cf. [1], 27A.7), and combining the above resultwith Theorem 3.1 we obtain the following: COROLLARY 4.3. Assume that conditions (E) and (S) hold. lf the functor Pt : X

P r T o p preserves finite products, then 7"2 = Sep. REMARKS 4.4. 1. Next we present some consequences of the above result. (a) In Example 2.3.5 conditions (S) and (E) are fulfilled and the functor Pt : Top c --~ 79rTop preserves finite products. Therefore, Sep = 72. (b) Now let us consider Example 2.3.7. Conditions (S) and (E) hold and it is easily verified that the functor Pt : Grph --+ 79rTop preserves finite products,

HAUSDORFF SEPARATION IN CATEGORIES

295

hence we also have that S e p = 7"2. 2. In the Corollary the preservation o f finite products is essential. For instance, if we consider the category T o p and the regular closure operator induced by the subcategory H r y o f Urysohn spaces (with respect to the class o f embeddings) and the 'traditional points' (i.e. the choice o f Example 2.3.1.(a)), then P t ( X ) is a topological space for e v e r y space X . However, S e p does not coincide with 7"2 " S e p is exactly H r y and T2 is (an extremally epireflective subcategory) properly contained in H r g (see [8] and [4]).

References 1. E. Cech: Topological @aces, revised by Z. Frol~ and M. Kat~tov, Academia, Praha, 1966. 2. M.M. Ctementino: Separa~o e Compacidade em Categofias, Ph.D. Thesis, Universidade de Coimbra, 1992. 3. J. de Vries: Topological Transformation Groups 1, Math. Centre Tracts 65, Mathematisch Centrum, Amsterdam, 1975. 4. D. Dikranjan andE. Giuli: Epimorphisms andcowellpoweredness ofepireflectivesubcategories of Top, Rend. Circ. Mat. Palermo, Ser. 116 (1986), 121-136. 5. D. Dikranjan and E. Giuli: Closure Operators I, TopologyAppl. 27 (1987), 129-143. 6. D. Dikranjan, E. Giuli, and W. Tholen: Closure Operators II, Categorical Topology and Its Relations to Analysis, Algebra and Comhinatorics, Proc. of the Prague Int. Conf. 1988 (World Cientific, Singapore-New Jersey-London-Hong Kong), 297-335. 7. L. Fuchs: Infinite Abelian Groups, Vol. I, Academic Press, New York-London, 1970. 8. E. Giuli and M. Hu~ek: A diagonal theorem for epireflective subcategories of Top and cowellpow6redness, Ann. Mat. Pura Appl. 145 (1986), 337-346. 9. E. Giuli, S. Mantovani, and W. Tholen: Objects with closed diagonals, J. Pure AppL Algebra 51 (1988), 129-140. 10. H. Herrlich, G. Salicrup, and G. E. Strecker: Factorizations, denseness, separation and relatively compact objects, TopologyAppl. 27 (1987), 157-169. 11. I.M. James: Fibrewise Topology, Cambridge University Press, Cambridge, New York, 1989. 12. E. Manes: Compact Hausdorff objects, TopologyAppl. 4 (t974), 341-360. 13. D. Pumpl0n and H. R0hrl: Separated totally convex spaces, Manusc. Math. 50 (1985), 145-183. 14. W. Tholen: Semi-topological functors I, J. PureAppI. Algebra 15 (1979), 53-73. 15. S. Willard: General Topology, Addison-Wesley Publishing Company Inc., 1970.