Advances in Pure Mathematics, 2011, 1, 284285 doi:10.4236/apm.2011.15051 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Property S[a,b]: A Direct Approach George Nickolaos Miliaras American University of Athens, Athens, Greece Email: gmiliara@yahoo.gr, gmiliaras@aua.edu Received May 18, 201 1; revised June 17, 2011; accepted June 30, 2011 Abstract In this paper we prove directly that the property ,Sab, implies ,ab compact, and under certain condi tions it implies compact. ,ab Keywords: Compactness number, compact, ,ab ,ab compact, property ,Sab 1. Introduction Compactness is one of the oldest and the most famous notions in mathematical analysis and especially in to pology. A partial generalization is ,ab compactness [18]. This has been shaped to property ,Sab by Vaughan in 1975 [8], (page 253 and 256257) who proved that ,ab compactness is equivalent to ,Sab if (Theorem 2C) using the Corol lary of Lemma 2 (pages 254255). b acf In this paper we are going to prove directly some thing stronger, which we will need the following defi nitions: Definition 1. The compactness number Cn X of a space is the least cardinal such that every open cover of k has a subcover of cardinality less than . k Definition 2. A space is called ,ab U compact ( compact) if every open cover of ,ab with Ub (Ub) has a subcover of cardinality strictly less than . a Definition 3. A space is said to have property ,Sab if every open cover of of regular cardi nality less than b, has a subcover of cardinality strictly less than a. 2. Main Result Theorem. Let have the property ,Sab, then is compact, and if is regular or if , then ,ab a b bcf is compact, furthermore either , or, . ],[ ba ab Cn Xb CXn Proof. We study the following three cases: Case 1. Cn Xb Let * be an open cover of U with * Uk be the first singular cardinal greater than , with the property that h as no subcover with cardinality less than . a * U a If , then clearly kb is ,ab compact. If kb , then clearly is compact. ,ab Assume that kb . Since , there exists an open cover of CnXkb * V which * Vk . Then at least one * UU is covered by a collection ** VV with * V k , such that no subcollection of * V with less than elements can cover U. If such a U didn't exist it would contradict the hy pothesis XCn bk. Consider the collection of open sets V * U * WV . Let a collection V * WW k of k elements of * W and let W be the union of the rest of the remaining elements of * (if there are any left). We have W * UW W , put ** UU U , then * is an open cover of W ** SUW with kS* since * Uk , * Wk and 1W, then * has a subcover with cardinality less than , since is a regu lar cardinal and S ak b has property ,bSa . Now, since * refines *, must have a subcover with car dinality less than , thus SU U* a is compact. ,ab Now, if is a regular cardinal then clearly b is ,ab compact, since it is compact, and has property ,ab ,Sab. Assume that is singular and . Let b cf ba * UU b , be an open cover of , with * Ub , let cfk ab , choose cardinals b , k with bb sup . For every k , let bVU , and let k * VV. Then * has a subcover * V with V * Va , since is regular, and k ka has property ,Sab . Let
G. N. MILIARAS285 * VV , then 1, but VX 11 VUb . Put *1 UU b , then *1 Ub b , then since is compact, if ,ab * Ua , it has a subcover * U with * Ua , thus is ,ab compact. Case 2. Cn Xb bAssume that is a limit cardinal. Since has property ,Sab, for every b , has property ,Sa , and therefore is a , compact for every b , and if is regular or , cf a is ,a compact, by case 1. Let * U be an open cover of , since Cn Xb , * has a subcover * such that U U* Ukb b . Now, since is a limit cardinal, , it follows from the above that b k is compact, so * U has a subcover * U,ak such that * Ua , thus is ,ab compact and since , we must have . Assume that is a successor cardinal , then Cn X b b CnX ab k has property , therefore ,k Sa has property ,Sak and therefore is compact. If is regular or , ,ak k k acf is ,ak compact, by case 1, and since Cn , Xk is compact, thus ,ak is ,ab compact, and since Cn Xb we have . CnX cf ab Assume that . Let * be an open cover of k aU with * Uk, with no subcover with cardinal ity less than . Let * be an open cover of k V with * Vk . Consider the open cover *** . Then ,UVV WW VUU* Wk and also refines *, since * W U has property , has a subcover * W with ,Sa k * W* Wa and since * W refines *, has a subcover * U U* U with * Ua , thus is ,ak compact and since , Cn Xkb is ,ab compact and using the previous argument XabCn. Case 3. Cn Xb Let X b Cn , then has property ,Sa , so Xb a Cn by case 2. Now since aCn X , is ,ac compact for every , therefore ca is ,ab compact. The proof is complete. 3. References [1] P. S. Alexandroff and P. Urysohn, “Memorie sub les Espaces Topologiques Compacts,” Koninklijke Akademie van Wetenschappen, Amsterdam, Vol. 14, 1929, pp. 196. [2] R. E. Hodel and J. E. Vaughan, “A Note on [a,b] Com pactness,” General Topology and Its Applications, Vol. 4, No. 2, 1974, pp. 179189. doi:10.1016/0016660X(74)900208 [3] G. Miliaras, “Cardinal Invariants and Covering Properties in Topolology,” Thesis, Iowa State University, Amster dam, 1988. [4] G. Miliaras, “Initially Compact and Related Spaces,” Periodica Mathematica Hungarica, Vol. 24, No. 3, 1992, pp. 135141. doi:10.1007/BF02330872 [5] G. Miliaras, “A Review in the Generalized Notian of Compactness,” Unione Matematica Italiana, Vol. 7, 1994, pp. 263270. [6] G. Miliaras and D. E. Sanderson, “Complementary Forms of [α, β]Compact,” Topology and Its Applications, Vol. 63, No. 1, 1995, pp. 119. doi:10.1016/01668641(95)900012 [7] R. M. Stephenson, “Initially kCompact and Related Spaces,” In: K. Kuuen and J. E. Vaughan, Eds., Hand book of SetTheoritic Topology, NorthHolland, Amster dam, 1984, pp. 603632. [8] J. E. Vaughan (Greensbaro N. C.), “Some Properties Related to [a,b]—Compactness,” Fundamenta Mathe maticae, Vol. 87, 1975, pp. 251260. Copyright © 2011 SciRes. APM
