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

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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 connected- ness, called sum connectedness, is introduced. The category of sum cohnected spaces forms the smallest co-reflective subcategory of TOP contaihing all con- nected 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 con- tinuous maps) containing all connected spaces. The other co-reflective subcate- gories 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 topo- logical 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 neigh- bourhood 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,.
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