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Carnegie Mellon University Research Showcase @ CMU Department of Mathematical Sciences Mellon College of Science 1968 Paracompact subspaces Richard A. Alo Carnegie Mellon University Harvey L. Shapiro Follow this and additional works at: http://repository.cmu.edu/math This Technical Report is brought to you for free and open access by the Mellon College of Science at Research Showcase @ CMU. It has been accepted for inclusion in Department of Mathematical Sciences by an authorized administrator of Research Showcase @ CMU. For more information, please contact [email protected]. NOTICE WARNING CONCERNING COPYRIGHT RESTRICTIONS: The copyright law of the United States (title 17, U.S. Code) governs the making of photocopies or other reproductions of copyrighted material. Any copying of this document without permission of its author may be prohibited by law. PARACOMPACT SUBSPACES Richard A. Alo and Harvey L. Shapiro Report 68-9 March, 1968 PARACOMPACT SUBSPACES by Richard A. Alo and Harvey L. Shapiro 1. Introduction. The definition of paracompactness for a topological space (X,3) states that every open cover of the space have a locally finite open refinement. space S When applying the definition to a sub- of (X,3), the phrases f open cover1 and ! open refinement1, of course, refer to the relative topology on also is in reference to open sets in S. S. Locally finite Immediate questions arise when these phrases are used in reference to the topology for the whole space S X. That is, one can consider a subspace of (X,3) to be STRONGLY PARACOMPACT if every cover of by members of members of ? 3. has a locally finite in X S refinement by (In [1], such subspaces were called a-paracompact.) It is clear that every strongly paracompact subspace of a topological space is paracompact. not equivalent. However, the two notions are This follows from the example of a completely regular, non-normal space due to Niemytzki. subset of the plane y >_ 0. (x,0) of Let R. S R x R Let X be that consisting of the points (x,y) for be that subset of X consisting of the points The relative product topology 3 on to include also as neighborhoods of the points in X S is enlarged the e-spheres tangent to the points (x,0) together with ((x,0)} for all G > 0. That is, the sets K(x,€) = { (x,0) U { (u,v)eS : (u - x ) 2 + (v - e ) 2 < e2}} are also included for all e > 0. With this new topology S is 2 a closed discrete subspace of However, D X and hence is paracompact. is not strongly paracompact. is an open in X cover of S For the cover (K(x,]/2)) that has no refinement and' is not locally finite in X. point (x,0) in will meet an infinite number of the sets K(x,l/2). S For, any 3 neighborhood N of a As our main theorem will show this subspace not strongly paracompact since it is not S P-embedded in is X. In particular, it will be shown that for normal Hausdorff spaces a subspace is strongly paracompact if and only if it is paracompact and topological space P-embedded. X A subset S is P-EMBEDDED in a if every continuous pseudometric on extends to a continuous pseudometric on X. S This concept was studied in [3] and [5]. Its relationship to paracompactness was studied in [4] and [6]. The notion of the notion of P-embedding generalizes C-rembedding where extensions of continuous real valued functions are required. The importance of P-embedding can be readily realized from the fact that it plays the same role in collectionwise normal spaces as C-embedding plays in normal spaces. In particular a space is collectionwise normal if and only if every closed subset is P-embedded (see [3]). Hence the notion of strongly paracompact, from the results in this paper, will have applications in such spaces. The notion of strongly paracompact can also be generalized by making use of infinite cardinal numbers 7. have already been applied to paracompactness and Such techniques P-embedding. Hence we will relate this new notion of 'strongly y-paracompact' y to the notions of 7-paracompact and P -embedded. Throughout this paper emphasis will be placed on the notions in the definitions of paracompactness that deal with the topologies of the subspace and of the space. Hence we will refer to, for example, the local finiteness of a cover as being either locally finite in S (the subspace) or locally finite in X. will mean a topological space with topology 2. Definitions and Basic Results. topological space (X,2) and let number. The relative topology on The subset 2 S is refinement by metnbers of y 3g. if every cover by members of a locally finite in special case of X y = /\ then and such that pseudometric topology if every be a subset of the be an infinite cardinal S will be denoted by has a locally finite in 3 of cardinality at most G 3^. 2. continuous extension to S S y has In the 7-paracompact is the usual notion G of S is dense in The subset S d on S is 7- whose cardinality is S is relative to the P -EMBEDDED in y-separable continuous pseudometric on The subspace 3 . The subset is STRONGLY y-PARACOMPACT A pseudometric SEPARABLE if there is a subset X «T. refinement by members of of countably paracompact. y S X 7-PARACOMPACT if every cover by members of of cardinality at most at most y Let Also (X,?) or S has a X. is PARACOMPACT (respectively STRONGLY PARACOMPACT) if every cover of ?) has a locally finite in S by members of S (respectively in ? o (respectively X) refinement 4 by members of 2 (respectively 3"). every continuous pseudometric on pseudometric on X. S It is P-EMBEDDED if extends to a continuous This is equivalent to saying that S is paracompact (respectively strongly paracompact, respectively P-embedded) if and only if S is y-paracompact (respectively y strongly y-paracompact, respectively cardinal numbers S y. P -embedded) for all By an extension is meant a cover of X U* of a cover such that the trace of U U* on on S is The definitions of other terms used in this paper can be found in [3]. The following results will be needed in the main part of this paper. Theorem 2.1 (see [3]). If y S is a subspace of X and if is an infinite cardinal number then the following statements are equivalent: S (2) Every locally finite in members of of P y -embedded in (1) 3 is 3^ of power at most X. S y normal cover of has a refinement that can be extended to a locally finite in cover of X (3) by members of members of 3 normal cover of ^ X by by members normal 3^. Every locally finite in by members of S S of power at most normal open cover of y has a refinement by that can be extended to a locally finite in X by members of S X 3. Theorem 2,2 (see for example [1]). If X is a topological space then the following statements are equivalent: 5 (1) X is normal. (2) Every locally finite open cover of Lemma 2.3 (Tukey, see [7]). Let by members of and let ^g of a subset lr = (Va) n S 3. n S C U for each a U = (U ) be a cover of a topological space V on S refines locally finite open cover to = (W ) a is normal. be a locally finite open cover of such that the trace of W X X. of X X Then there is a X such that aeIm Main Results. Theorem 3,1. If X is a topological space and if S is S is y a normal y-paracompact strongly y-paracompact. Proof. Le*t U* cardinality at most U of U* on S P -embedded subset of be a cover of y. Since S S is X9 then by members of 3> with y-paracompact, the trace has a locally finite in S refinement \s by members of ^ c . By Lemma 2.3, \s can be assumed to have cardinality at most y. Normality of S implies that \s is normal in S b y Theorem 2.2. Moreover, since S is Pyembedded in X, Theorem 2.1 states that extends to a locally finite in members of 3. X topology Hence on X. Theorem 3.2. S refinement of S has a refinement that normal cover Now the family {W PI V : a locally finite in 3 X V WeU>* U is strongly U>* of and X by VeU*} is by members of the y-paracompact. If (X,3) is a normal topological space and if is a closed subset of X, then the following statements are equivalent: y (1) (2) S S is y-paracompact and P -embedded in is strongly y-paracompact. proof« 3.1. S S is normal, (2) follows from (1) by Theorem On the other hand, if is y-paracompact. y is S Since P -embedded in cover of S then a cover at most y S is strongly X. In fact if of S U 3g of cardinality at most by members of By (2) , there is a locally finite in U*. 3 by members of 3 locally finite in finite in S X to of to* on X S G y to* of U 0 SeU. S such that to* by refines S. G = to* U (XNS} is a locally The family X by members of X 3 to U It is clear that cover of whose trace with is normal, G to is and S is is normal. P - X. Corollary 3.3. space and if cover for each is the refinement of is the required extension of embedded in Ue^ needed in 2.1. is to. By 2.2 and the fact that Hence y «T with cardinality with cardinality at most Then the trace S is a locally finite in is obtained by selecting one members of y-paracompact then Theorem 2.1 is now used to show that by members of U* X. S If X is a normal Hausdorff topological is a subset of X, then the following statements are equivalent: (1) S is strongly paracompact. (2) S is paracompact and Proof. X. In [1 , Corollary 4]it is shown that a strongly paracompact subset is closed. (2) # P-embedded in Thus 3.2 shows that (1) implies Since paracompact Hausdorff spaces are normal, Theorem 3.1 shows that (2) implies (1). This completes the proof. 7 Corollary 3,4, If is a topological space and if S is a paracompact Hausdorff P-embedded subset of S is strongly paracompact in X, Corollary 3,5, X is a topological space and if S v/ is a normal Hausdorff countably paracompact P -embedded subset of X 5 then S If X, then X is strongly countably paracompact. In [2] , it was shown that ja normal space X jjs 7-collectionwise y normal if and only if every closed subset is P -embedded in X. In particular^ it was shown that X jus normal if and only if PWo -embedded in every closed subset is X. Using these results it is now possible to obtain the following three corollaries. Corollary 3,6, If S is a closed subset of a normal 7-collectionwise normal space if and only if ' S is strongly Corollary 3,7. space X then X, then S If S S is 7-paracompact 7-paracompact. is a closed subset of a normal is countably paracompact if and only if S is strongly countably paracompact. Corollary 3,8, normal space If X, then S S is a closed subset of a collectionwise is paracompact if and only if S is S and strongly paracompact. Suppose S F is a strongly is a subset of a topological 7-paracompact subset of X. give some sufficient conditions for paracompact in It is now possible to F to be strongly 7- X, Theorem 3,9, Let F be a strongly 7-paracompact subset of the closed subspace S of the topological space X. If 8 there is an open set is strongly Let U* cardinality at most F of S y. G members of 3 3^. U* S in X F U of U* U F. S From X. G on S 3 is a cover has a locally #c. The trace \s refinement of c S Clearly with and V G U* by open in X is locally finite is closed. If of the closed subspace in then by members of by members of is open in Corollary 3.10. G F By assumption and covers since F c G ^ S X. is a locally finite in it follows that V X such that The trace refinement \s* on in X be a cover of by members of finite in in y-paracompact in Proof. of G such that F S F c is a strongly paracompact subset of (X,,3) and if there is an open set G c S then F is strongly paracompact X. Corollary 3.11. If F is a paracompact Hausdorff embedded subset of the closed subspace is an open set G in X strongly paracompact in Proof. such that F ^ G c: s then F is X. S. strongly paracompact in Hence by Corollary 3.10 X and if F S is is a closed paracompact is a closed subset of contained in the interior of X. F Y. Corollary 3.12 (see [1]). If compact in of (X,3) and if there By Corollary 3.4, F is a strongly paracompact subset of the closed subspace subset of S P- M, then F X that is is strongly para- 9 REFERENCES 1. C. E. Aull, Paracompact Subsets, Proc. of the Second Prague Topological Symposium, 1966. 2. T. E. Gantner, Extensions of Pseudometrics and Uniformities, Thesis, Purdue University, 1966. 3. H. L. Shapiro, Extensions of Pseudometrics, Canad. J. Math., 18 (1966), 981-998. 4. , Closed Maps and Paracompact Spaces, Canad. J. Math,, to appear. 5. , A Note on Extending Uniformly Continuous Pseudometrics, Bulletin de la Societe Mathematigue de Belqicrue, 18 (1966), 439-441. 6. , A Note on Closed Maps and Paracompact Spaces, to appear. 7. J. W. Tukey, Convergence and Uniformity in Topology, Princeton, Princeton Press, 1940. Carnegie-Mellon University Pittsburgh, Pennsylvania 15213