A Class of Spaces Containing All Connected and All Locally Connected Spaces

Math. Nachr. 82, 121-120 (1078)

A Class of Spaces Containing all Connected and all Locally Connected Spaces

By J. K. KOHLIof Delhi (Inditi)

(Eingegangen am 8. 12.1975)

Abstract. A simultaneous generalization of connectedness and local connectedness, called sum connectedness, is introduced. The category of sum cohnected spaces forms the smallest co-reflective subcategory of TOP contaihing all connected spaces. A product theorem, analogous to that for locally connected spaces, is proved for sum connected spaces and a necessary and sufficient condition for the STONE-CECHcompactificetion of a TYCHONOFFspace to be sum connected is obtained.

8 1. Introduction

We say that a topological space X is sum connected if each component of LY is open. The class of sum connected spaces properly includes the class of connected spaces and the class of locally connected spaces; and forms the smallest co- reflective subcategory of TOP ( =the category of topological spaces and continuous maps) containing all connected spaces. The other co-reflective subcategories of TOP which have attracted wide attention and have been studied exten- eively are the categories of sequential spaces [6], k-spaces [l], chain net spaces [l 11 locally (pathwise) connected spaces ([8], [9]), and P-spaces [14]. In 5 2. we introduce a localization of sum connectedness and study the topological properties of sum connected spaces. The preservation of topological properties under the corresponding co-reflective functor is studied in Q 3. In $4. we obtain a necessary and sufficient condition for the STONE-~ECHcompzcti- fication of a TYCHONOFFspace to be sum connected. Some examples are included in Q 5. Sum connected spaces are also related to the natural covers of FRANKLIN[7] and BROWN[3]. MICHAEL’Stheorem on the existence of selections is closely related to sum connected spaces [13]. Throughout the paper, the symbols N and I will denote the space of natural numbers and the closed unit ihterval, respectively, endowed with the usual 122 KOHLI,A Class of Spaces topologies. By a clopen set we mean a set which is both open and closed. The closure of a set A in a space X will be denoted by cl,A or A. For any TYCHONOFF space X,,9X will denote its STONE-CECRcompactification.

8 2. Localization and Topological Properties

2.1. Definitions. Let X be a topological space and let xEX. We say that X is (i) sum connected at x if there exists an open connected neighbourhood of x; (ii) weakly sum connected at x if there is a connected neighbourhood of x (or equivalently the component containing x is a neighbourhood of x); (iii) quasi sum connected at x if the quasi component containing x is a neighbourhood of x; (iv) padded at x if for every neighbourhood U of x, there are open sets W and V such that xE W&Ws VC, U,and V\W has only finitely many components. A space X is said to have any of the properties defined above if it has the property at each of its points. 2.2. Proposition. If X is sum connected at x, then X is weakly sum connected at x. If X is weakly sum connected at x, then X is quasi sum connected at x. Neither of these implications can be reversed. Proof. First two implications are immediate in view of definitions. For the last part see Examples 5.2 and 5.3. 2.3. Proposition. If X is a dense subspace of Y and if X is sum connected at x, then Y is sum connected at x. Proof. Let U'= UnX be an open connected neighbourhood of x in X,where U iu an open neighbourhood of x in Y. Since X is dense in Y,UsclyU'= = cl, (Un X)and so U is an open connected neighbourhood of x in Y. In contrast to connectedness, the closure of a sum connected set need not be sum connected even in a sum 'cohnected space. 2.4. Theorem. The following conditions on a topological space X are equivalent. (a) X is quasi sum connected. (b) X is weakly sum connected. (c) X is sum connected. (d) (quasi)components of X are open. (e) X has weak topologyl) with respect to its connected subsets. Proof. Since an open quasi-component is a component, the two assertions in (d) are equivalent. Again, since a set is open if and only if it contains a neighbourhood of each of its points, (d) is equivalent to (a), (b), and (c). The equi- valence of (d) and (e) follows because a conhected set intersects a component C if and only if it is cohtaihed in C.

1) See FRANKLIN[7] for the definition. KOHLI,A Class of Spaces 123

2.5. Corollary,A space is sum connected if and only if it is the disjoint topological sum of its (quasi)components2). 2.6. Corollary. A function on a sum connected space is continuous if and only if it is continuous on each component. 2.7. Corollary. A locally connected space is the disjoint topological sum of its (quasi)components. 2.8. Corollary. In a sum connected space components and quasi-components coincide. In particular, components and quasi-components coincide on every open subset of a locally connected space. Proof. The first assertion follows from 2.4(d). For the last assertion we need only note that every locally connected spce is sum connected and that local connectedness is open hereditary. The following proposition extends a result of DEGROOTand MCDOWELL[5]. Although, the proof is similar in general case, we include it for the sake of com- pleteness. 2.9. Proposition. If a sum connected space X is padded at x, then X is locally connected at x. Proof. Let U be an open neighbourhood of x. Since X is sum connected, we may assume that U is contained in a component C of X. Choose open neigh- bourhoods W and V of x such that vsVz U and V\w has only finitely many Components C,, C,, . . . , C,. For each i, 1 sisn, there is a quasi-component Qi of V such that C,&Qi (the Qi need not be distinct). We assert that each vE V is in some Qi. If not, for each i snthere is a set Vi, clopen in V,containing v and n missing Qi. But then n Viis open in V and closed in W,and is therefore clopen i=l in C, which is a contradiction to the fact that C is a component of X. Since V has only finitely many quasi-components, each of them is open, and therefore a component. Thus the component of x in V is an open connected neighbourhood of x lying in U. DEGROOTand MCDOWELLhave given an example of a connected space x' such that X is locally connected at a point xEX but fails to be padded at x ([5], Example 5.1). Thus the converse of the above proposition is false. 2.10. Proposition. Every quotient of a sum connected space is sum connected. Proof. Let f be a quotient map of a sum connected space X onto Y. Let C be any component of Y and let z€f-l(C).Then x lies in a component C, of X. Thus f(C,) is connected and f(x)Ef(C,)nC.Therefore, f(C,)sC and consequently, C,cf-'(f(C,))sf-'(C). Since X is sum connected, C, is open. Thus f-l(C) being a neighbourhood of each of its points is open in X. Hence C is open in Y.

2) This characterization led Dr. A. K. CHILANA to suggest the present nomenclature for which author wishes to thank her. 124 KOHLI,A Class of Spaces

2.11. Corollary. Continuous open or closed imuges, adjunctions or inductive limits of sum connected spaces are sum connected. 2.12. Corollary. If a product space is sum connected, so is each of its factors. In view of 2.4(d) it follows that the disjoint topological sum of any family of sum connected spaces is sum connected. Thus a characterization of co-reflective subcategories of TOP ([12], Theorem 6) together with 2.10 yields the following, if one notes that co-products and extremal quotient objects in TOP are disjoint topological sums and topological quotients, respectively. 2.13. Theorem. The full subcategory of sum connected spaces is the smallest co-reflective subcategory of TOP containing all connected spaces. 2.14. Remark. In view of 2.4(e) an alternative proof of 2.10 and 2.13 can be given using the theory of natural covers ([3], [7]). Infact, sum connected spaces are precisely those that one obtains by assigning to a space the natural cover consisting of all connected subsets. In general the property of being sum connected is not productive. For example, any infinite product of a two point discrete space is not sum connected. However, we have the following. 2.15. Proposition. A product of sum connected spaces is sum connected if and only if all but finitely many factors are connected. Proof. Let X=nX, be the product of a family {Xu}of sum connected spaces such that all but finitely many Xuare connected and let C be a component of X. Then p,(C) is connected for each a, where pe danotes the projection into a-th co-ordinate. Thus the product np,(C) is connected. We claim that npu(C)=C. For each a, let C, denote the component of Xu which contains pu(C)and let C’=ltC,. Now, if p,(C) is a proper subset of C, for some a, then C is a proper subset of C’. Therefore, Cu=pu(C)for each u. Moreover, by the hypothesis on the spaces Xu,pu(C) =Xu for all but finitely many a. Thus C is a basic open set in X. Conversely, suppose X=nX, is sum connected. By 2.12 each Xu is sum connected. Moreover, if X, is not connected for all but finitely many a, then no component of X is open. This completes the proof. Simple examples can be given to show that sum connectedness is neither open hereditary nor closed hereditary. Infact, a space X is open hereditarily sum connected if and only if X is locally connected. 2.16. Proposition. A pseudocompact sum connected space has at most finitely many components. Proof. Let X be a pseudocompact sum connected space. Suppose X has infinitely many components.- Choose an infinite sequence {Ci} of distinct components of X and let C= u Ci.Definef: X-R by f(x)=O if xcX\C and f(x)=n i= I if XCC,, n= 1,2, . . . Then f is an unboufided cofitinuous real-valued function contradicting the pseudocompactness of X. We omit the proof of the following proposition. KOHLI,A Class of Spaces 125

2.17. Proposition. A LINDELOF(or separable) sum connected space has at most countably many components. 2.18. Remark. If we call a space X sum pathwise connected in case each path component of X is open, then essentially all the results of this section are also true for sum pathwise connected spaces. Moreover, sum pathwise connected space are precisely thoee that are obtained by assigning to each space the natural cover consisting of all pathwise connected subsets.

8 3. Coreflective Functor

In view of the Theorem 2.13 one may ask: Which topological properties are preserved under the corresponding co-reflect'ive functor ? This section is devoted to answer this question. The analogous problem for the locally connected functor has been studied by GLEASON[9]. Let (X, T) be a topological space and let 9 denote the union of T and the collection of all T-components of X. Let T* denote the topology generated by takingd as D subbase. 3.1. Proposition. If (X,T) is first countable, so is (X,T*). Proof. Let {V,},,, be a countable T-baseat xEX. Let C be the T-component containing x. Then (Vn}nENwhere V,,=Cfl U, is a countable T*-base at x. 3.2. Proposition. If Y is sum connected and if f is a function from Y into X continuous relative to T, then f is also T*-continuous. Proof. Let Sf$. If SET, then since f is T-continuous, f-l(S) is open in Y. If S $ T, then 5' is a T-component of X. Let pE f-l(s)and let C, be the component of p in f-'(S). Since f is T-continuous, C, is a component of Y. Since Y is sum connected, C, is open. Thus f-'(S) being a neighbourhood of each of its points is open. 3.3. Proposition. The spaces (X,T) and (X, T*) have same path components. Proof. By 3.2 any function f: I--X which is T-continuous is also T*-continuous. 3.4. Proposition. The collection T* is the weakest sum connected topology for X containing T. Proof. First we show that (X,T*) is sum connected. Let C be a T*-component of X. Since T 5 T*, C is T-connected. We assert that C is a T-component of X. For if not, then C is a proper subset of a T-component C' of X; but any T*- partition of C 'induces a T-partition of C' and thus leads to a contradiction. Since TGd, T* is stronger than T. Suppose T' is a sum connected topology for X containing T. Then the identity map 1, of Y=(X, T') onto (X,T) is continuous and hence by Proposition 3.2 is a continuous map of Y onto (X,T*). Thus T' is stronger than T*. 126 KOHLI,A Class of Spaces

3.5. Proposition. The ,spaces (X, T) and (X, T*) have same connected subsets. Proof. By the construction of T*, it follows that for any T-connected subset C of X, the tyo relative topologies TIC and T*/C on C coincide. We conjecture that T* is the largest sum connected topology on X having same connected subsets as T. In view of Propositions 3.4 and 3.5 it follows that (X, T*) is honieomorphic to the disjoint topological sum of T-components of X. Hence any topological property which is closed hereditary and is preserved under disjoint topological sums will be preserved in the passage from (X, T) to (X, T*). Properties which are closed hereditary and are preserved under disjoint topological sums include the following : (a) Separation properties. For example, To, Ti, T,, T:], T i . 3T (b) Forms of normality. For example, normality, hereditary normality, perfect normality, collectionwise normality, full normality ( = paracompactness). (c) metrizability, local compactness and local weight. However, the simple example of CANTORset shows that the compactness, (countable, sequential or pseudo) compactness, second countability and separa- bility need not be preRerved in the passage from T to T*. 3.6. Proposition. If (X, T) is a topological group, so is (X,T*). Proof. It is immediate in view of Propositions 3.4, 2.15 and 3.2. We shall denote the n-th homotopy groups of (X, T) and (X, T*) based at xoEX by nn(X,x,,) and nn(sX,x,,), respectively. 3.7. Proposition, The spaces (X, T) and (X,T*) have same homotopy groups. In particular, (X, T) and (X,T*) have same fundamental groups. Proof. In view of Proposition 3.2, it follows that there is one-to-one corre- spondence between n-dimensional hyperloops of (X,T) and (X,T*). Again, iff and g are two n-dimensiohal hyperloops in (X, T) which are homotopic, then f and g are also homotopic as hyperloops in (X, T*). Thus n,(X, xo)and n,(sX, xg) are isomorphic for each zoCX. The last assertion follows because the fundamental group of a space is precisely the l-dimensional homotopy group. 3.8. Proposition, The spaces (X,T) and (X,T*) have same singular homology (cohomology ) groups. Proof. This follows from Propositions 3.3, 3.5, and ([15], p 175 and 239).

g 4. Sum Connected Compactifications

Local connectedness of STONE-~ECHcompactifications has been investigated by BANASCREWSKI[2], DEGROOTand MCDOWELL[5], HENRIRSENand ISBELL[lo] and others. In this section we investigate under what conditions a completely regular HAUSDORFFsum connected space admits a sum connected HAUSDORFF KOHLI,A Class of Spaces 127

compactification. Throughout the section all spaces are assumed to be completely regular and HAUSDORFF. 4.1. Proposition. If X is sum connected and has only finitely many components, then ,6X is sum connected. In particular, if X is sum connected and pseudocompact, then PX is slim connected. n n Proof. Let X = u Ci,where each Ciis a component of X. Then PX = u clp,Ci ,=i i=l and each clpxCi is connected in PX. Since each Ci is clopen in X, each clpsCi is clopen in j3X, and hence is a component of PX. Thus PX is sum connected. The last assertion follows from Proposition 2.16. 4.2. Proposition. If PX is sum connected, then X is sum connected and has finitely many components. Proof. By Proposition 2.16, /?X has finitely many components. Say, PX= n = u Ci,where each Ciis a component of PX and hence clopen in PX. Then i=i n X= U (CinX).Each C,nX is noneniptp and clopen in X. We assert that each i=l GinX is connected in X. For otherwise, since X is dense in PX and since Cifl X is open in X,Ci=clpx(CinX), and consequently any partition of CinX will induce a partition of Ci in PX, contradicting the fact that Ciis a Component of PX. Therefore, each C,nX is a component of X. Thus X is sum connected and has only finitely many components. 4.3. Corollary. If X is sum connected and has only finitely many components, then every compactification of X is sum connected. Proof. Any HAUSDORBFcompactification bX of X is a continuous image of PX. Thus bX is a continuous closed image of PX and hence sum connected by 4.2 and 2.1 1. 4.4. Corollary. If X is sum connected and pseudocompact, then every compactification of X is sum connected. The following theorem sums up the results in 4.1 and 4.2. 4.5. Theorem. For a completely regular HAUSDORFFspace X the following statements are equivalent. (a) PX is sum connected. (b)If XsTSPX, then T is sum connected, i.e., every space in which X is dense and C*-embedded is sum connected. (c) X is sum connected and has only finitely many components.

5 6. Examples

6.1. Example. Let Y be the closure of the topologist’s sine curve. That is, Y = ((0,y) : - 1 sy 11 .( (z,sin t): o-=x 1) . 128 KOHLI,A Class of Spaces

Let X denote the disjoint topological sum of two copies of Y.Then X is a compact sum connected space which is neither connected nor locally connected. Further, a multiplication in X may be defined so that X becomes a topological semigroup. We do not know an example of a sum connected topological group which is neither connected nor locally connected. A space is sum connected if and only if its component space (the space of components with quotient topology) is discrete. BROWN[a] has shown that if x is a sum connected space with a base point, then the free topological group F(X) is sum connected and that the component space of a sum connected projective topological group is a discrete, free group. It seems interesting to study the free topological group on a sum connected group which is neither connected nor locally connected:’). 6.2. Example. In the Euclidean plane, let B, be the infinite broom consisting of the closed line segments joining the point - L): m=n,n+1, . . . , n=l,2,. . . Let B= u B,. m n=i Further, let A consist-of the closed interval [0,2] on the z-axis together with a sequence of closed intervals [l, 21 x {:I - , n= 1,2, . . . and let X =Au B. Then no open set containing (0, 0) is connected, so X is not sum connected at (0, 0). However, the component containing (0,O) contains all the B, and the interval [0, 21 on the x-axis. Hence X is weakly sum connected at (0, 0). 5.3. Example. Consider the sequence of poinb in the Euclidean plane xn= 1 =(I-;, I-;), n=l, 2,. . .For each n, let S, be thesquare with x, and x,+~

at opposite__ ends of a diagonal, and let R, be the rectangle two of whose sides are sides of S, and Sn+i,and whose fourth vertex is The space X consists of the point p = (1, l),together with a countable collection of closed intervals chosen as follows. In each 8, (each R,) choose a sequence of segments with end points on opposite sides ofS,(R,) converging to the line com- mon to 8, and R, (Rkand Sk+i).Then X is quasi sum connected at p but the component containing p.is a singleton.

References

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3) This problem was suggested by Prof. R. BROWNin a letter to the author. KOHLI,A Class of Spaces 129

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Hindu College University of Delhi Delhi -110007, India

9 Mnth. Nachr. Bd. 82