 Advances in Pure Mathematics, 2011, 1, 284-285 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 E-mail: 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 [1-8]. This has been shaped to property ,Sab by Vaughan in 1975 , (page 253 and 256-257) who proved that ,ab -compactness is equivalent to ,Sab if (Theorem 2C) using the Corol-lary of Lemma 2 (pages 254-255). b acfIn 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 X is the least cardinal such that every open cover of kX has a subcover of cardinality less than . kDefinition 2. A space X is called ,abU-compact (-compact) if every open cover of ,abX with Ub (Ub) has a subcover of cardinality strictly less than . aDefinition 3. A space X is said to have property ,Sab if every open cover of X of regular cardi-nality less than b, has a subcover of cardinality strictly less than a. 2. Main Result Theorem. Let X have the property ,Sab, then X is -compact, and if is regular or if , then ,ababbcfX is -compact, furthermore either , or, . ],[ baabCn XbCXnProof. We study the following three cases: Case 1. Cn XbLet * be an open cover of UX with *Uk be the first singular cardinal greater than , with the property that h as no subcover with cardinality less than . a*UaIf , then clearly kbX is ,ab -compact. If kb, then clearly X is -compact. ,abAssume that kb. Since , there exists an open cover of CnXkb*VX 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 bk. 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X with kS* since *Uk, *Wk and 1W, then * has a subcover with cardinality less than , since is a regu-lar cardinal and Sak bX has property ,bSa . Now, since * refines *, must have a subcover with car-dinality less than , thus SU U*aX is -compact. ,abNow, if is a regular cardinal then clearly bX 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 X, 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 kkaX has property ,Sab . Let G. N. MILIARAS285 *VV, then 1, but VX11VUb. Put  *1UU b, then *1Ubb, then since X is -compact, if ,ab*Ua, it has a subcover *U with *Ua, thus X is ,ab -compact. Case 2. Cn XbbAssume that is a limit cardinal. Since X has property ,Sab, for every b, X has property ,Sa, and therefore X is a,-compact for every b, and if  is regular or , cfaX is ,a-compact, by case 1. Let *U be an open cover of X, since Cn Xb, * has a subcover * such that UU*Ukbb. Now, since is a limit cardinal, , it follows from the above that bkX is -compact, so *U has a subcover *U,ak  such that *Ua , thus X is ,ab -compact and since , we must have . Assume that is a successor cardinal , then Cn XbbCnXabkX has property , therefore ,kSa X has property ,Sak and therefore X is -compact. If is regular or , ,ak kk acfX is ,ak -compact, by case 1, and since Cn , Xk X is -compact, thus ,akX is ,ab -compact, and since Cn Xb we have . CnXcfabAssume that . Let * be an open cover of k aUX with *Uk, with no subcover with cardinal-ity less than . Let * be an open cover of k VX with *Vk. Consider the open cover ***. Then ,UVV WW VUU*Wk and also refines *, since *W UX has property , has a subcover *W with ,Sak *W*Wa and since *W refines *, has a subcover *UU*U with *Ua, thus X is ,ak -compact and since , Cn XkbX is ,ab -compact and using the previous argument XabCn. Case 3. Cn Xb Let X bCn, then X has property ,Sa, so Xb aCn by case 2. Now since aCn X, X is ,ac -compact for every , therefore caX is ,ab -compact. The proof is complete. 3. References  P. S. Alexandroff and P. Urysohn, “Memorie sub les Espaces Topologiques Compacts,” Koninklijke Akademie van Wetenschappen, Amsterdam, Vol. 14, 1929, pp. 1-96.  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. 179-189. doi:10.1016/0016-660X(74)90020-8  G. Miliaras, “Cardinal Invariants and Covering Properties in Topolology,” Thesis, Iowa State University, Amster-dam, 1988.  G. 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