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Volume 38, 2011 Pages 301β308 http://topology.auburn.edu/tp/ Almost H-Closed by Jack Porter Electronically published on November 5, 2010 Topology Proceedings Web: Mail: E-mail: ISSN: http://topology.auburn.edu/tp/ Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA [email protected] 0146-4124 c by Topology Proceedings. All rights reserved. COPYRIGHT β http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 38 (2011) Pages 301-308 E-Published on November 5, 2010 ALMOST H-CLOSED JACK PORTER Abstract. This paper develops an example of a non-H-closed, Urysohn space with the property that for every chain of nonempty sets, the intersection of the π-closures of the sets is nonempty, every inο¬nite set has a π-complete accumulation point, and every π -sequence has a π-cluster point for every inο¬nite cardinal π . This shows that three of the characterizations of compactness cannot be extended directly to H-closed spaces. 1. Introduction In 1924, Paul Alexandroο¬ and Paul Urysohn [1] established a number of characterizations of compactness including these two: a space is compact if and only if every chain of nonempty closed sets has nonempty intersection if and only if every inο¬nite subset has a complete accumulation point. Also in [1], Alexandroο¬ and Urysohn introduced and characterized the concept of H-closed spaces. The chain characterization plays an important role in proving spaces are compact; for example, one major result connecting compactness and H-closure is that a space is compact if every closed subset is H-closed. Both M. H. Stone [9] and Miroslav KateΜtov [4] prove this result by showing that every chain of nonempty H-closed sets has nonempty intersection. 2000 Mathematics Subject Classiο¬cation. 54D25, 54D30. Key words and phrases. H-closed spaces, π -nets, π -sequences, π-closure, πcluster. c β2010 Topology Proceedings. 301 302 J. PORTER There are two natural paths to extend the chain condition to characterize H-closed spaces. Both extended conditions are satisο¬ed in H-closed spaces. This paper develops a relatively simple example that shows neither extended condition is enough to prove H-closure. In [1], Alexandroο¬ and Urysohn extended the complete accumulation point characterization of compactness to H-closure by proving that for each inο¬nite subset π΄ of an H-closed space π, there is a point π β π such that for each neighborhood π of π, β£π΄ β© πππ π β£ = β£π΄β£. In 1960, G. A. Kirtadze [5] showed that the converse is not true. Our simple example also shows the converse is not true. Recently, in an excellent paper about π -nets, R. E. Hodel [3] characterized H-closed spaces in terms of π -nets. Hodel asks if a Hausdorο¬ space π remains H-closed if π -nets in his H-closed characterization are replaced by a weaker notion of π -sequence. We show this is not true, too. All spaces considered in this paper are Hausdorο¬, and for a space π, let π (π) denote the set of open subsets of π. 2. Example 1 Recall (see [8, 4.8(e)]) that in a space π, a subset π΄ β π is regular closed if π΄ = πππ πππ‘π π΄. In 1940, KateΜtov [4] proved that a Hausdorο¬ space π is H-closed if and only if every ο¬lter base of regular closed subsets has nonempty intersection. One natural path of extending the chain condition to H-closed spaces would be to determine if a Hausdorο¬ space is H-closed if every chain of nonempty regular closed sets has nonempty intersection. We show this is false with an example of a non-H-closed, Tychonoο¬, extremally disconnected space πΉ. We need some preliminary results. Lemma 1. Let π½π be the Stone-CΜech compactiο¬cation of the countable discrete space π. (a) ([2, 9H.2]) If π΄ is an inο¬nite closed subset of π½π, then β£π΄β£ = β£π½πβ£ = 2π . (b) ([2, 6S.8]) If π΄ is a nonempty GπΏ subset of π½πβπ, then πππ‘π½πβπ π΄ β= β . Lemma 2. Let π be a chain of inο¬nite clopen sets of π½π. Then β£ β© πβ£ = 2π . ALMOST H-CLOSED 303 Proof: There is an ordinal π½ and a coο¬nal, well-ordered subchain π β² = {πΆπΌ : πΌ < π½} β π such that πΆπΌ β πΆπΌ+1 for πΌ + 1 < π½. As π β² is coο¬nal, β©π = β©π β² . If π½ is not a limit ordinal, then πΆπ½β1 = β©π β² , β£πΆπ½β1 β£ = 2π , and we are done. So, we can assume that π½ is a limit ordinal. Note that for πΌ < π½, πΆπΌ = πππ½π (πΆπΌ β© π) β πΆπΌ+1 = πππ½π (πΆπΌ+1 β© π). It follows that πΆπΌ β© π β πΆπΌ+1 β© π. Thus, {(πΆπΌ βπΆπΌ+1 ) β© π : πΌ < π½} is a family of pairwise disjoint subsets of π. So, π½ is a countable limit ordinal and ππ (π½) = π. There is a coο¬nal subfamily π = {π·π : π β π} of π β² . But β©{π·π β{π} : π β π} is a GπΏ set in π½πβπ and has nonempty interior. Hence, β£ β© {π·π βπ : β‘ π β π}β£ = 2π . Fact 3. Let π β π½πβπ and πΉ = π½πβ{π}. If π is a chain of nonempty regular closed sets in πΉ, then β©π = β β . Proof: First, note that πΉ is extremally disconnected as πΉ is a dense subspace of the extremally disconnected space π½π. Thus, the regular closed subsets of πΉ are clopen sets. If πΆ β π, then πΆ = πππΉ (πΆ β© π) = πππ½π (πΆ β© π)β{π} and πππ½π (πΆ β© π) = πππ½π πΆ is clopen in π½π. Also, β©π = β©{πππ½π πΆ : πΆ β π}β{π}. By Lemma 2, β£ β© {πππ½π πΆ : πΆ β π}β£ = 2π . Thus, β©π = β β . β‘ We have shown that the Tychonoο¬, extremally disconnected space πΉ has the property that every chain of nonempty regular closed sets has nonempty intersection. Since a regular H-closed space is compact (see [8, 4.8(c)]), πΉ is not H-closed. 3. Example 2 For π΄ β π, let πππ π΄ = {π β π : for π β π β π (π), πππ π β© π΄ β= β }; if there is the possibility of confusion, we will denote πππ π΄ by ππππ π΄. In 1968, N. V. VelicΜko [10] proved β© that a space is H-closed if and only if for each ο¬lter base β±, πΉ ββ± πππ πΉ β= β . In this section, we develop a Hausdorο¬, non-H-closed space β© π with the property that for every chain π of nonempty sets, πΆβπ πππ πΆ β= β . Surprisingly, the space π has other nice properties and answers a question by Hodel [3]. (1) The description of our space starts with a variation of the Tychonoο¬ plank. Let π denote the Tychonoο¬ plank (π1 + 1) × (π + 1)β{(π1 , π)} β the product space without the corner point; π is a zero-dimensional, Tychonoο¬ space. Let π = {(π1 , π) : π β π} be 304 J. PORTER the right-hand edge and π = {(πΌ, π) : πΌ β π1 } be the upper edge. Sometimes, we will need to consider a one-point extension of π, namely π βͺ {π}. The subspace π is dense and open in π βͺ {π} and {ππ½,π : π½ β π1 , π β π} is an open neighborhood base at the point π where ππ½,π (π) = {π} βͺ (π½, π1 ) × (π, π). (Note that this is the same space as enlarging the product topology (π1 +1)×(π +1) by adding the set {(π1 , π)} βͺ π1 × π.) The extension π βͺ {π} is H-closed (use 4.8(h)(8) in [8]) and Urysohn but not compact. Note that for πΌ < π1 , the subspace (πΌ + 1) × (π + 1) of π is compact. uπ (π½,rπ) π ππ½,π (π) Space π βͺ {π} e r(π1 , π) π (2) Let π denote π ×{0, 1, 2} ({0, 1, 2} has the discrete topology) with these points identiο¬ed: (π1 , π, 0) = (π1 , π, 1) for each π β π and (π½, π, 1) = (π½, π, 2) for π½ β π1 . For π β {0, 1, 2}, let ππ = π×{π}, ππ = π ×{π}, and π π = π ×{π}. Finally, let π = π βͺ{π0 , π2 }. For π β {0, 2}, an open neighborhood base for ππ is {ππ½,π (ππ ) × {π} : π½ β π1 , π β π}. Note that π0 βͺ{π0 } and π2 βͺ{π2 } are homeomorphic to π βͺ{π} and are H-closed subspaces of π and that πππ (π0 βͺ{π0 }) = π0 βͺ {π0 } and πππ (π2 βͺ {π2 }) = π2 βͺ {π2 }. π2 Space π π0 u π0 π2 u e (π½, π, 1) =r (π½, π, 2) r π1 (π1 , π, 0) = (π1 , π, 1) ALMOST H-CLOSED 305 Property A. The space π has the property that for every chain π of nonempty sets, the chain {πππ πΆ : πΆ β π} has nonempty intersection. Proof: β© Assume there is a chain π of nonempty sets in π such that {πππ πΆ : πΆ β π} = β . Let π denote the H-closed subspace π0 βͺ {π0 }. If πΆ β© π β= β for each πΆ β π, then π0 = {πΆ β© π : πΆ β π} is a chain of nonempty sets in the H-closed subspaceβ©π . As π0 is a ο¬lter base on π , it follows from VelicΜkoβs result that {ππππ (πΆ β© π ) : πΆ β π} = β β . However, ππππ (πΆ β© π ) = ππππ (πΆ β© π ) in π. This is a β© contradiction as {ππππ πΆ : πΆ β π} = β . So, there is a πΆ β π such that πΆ β©(π0 βͺ{π0 }) = β . A similar argument yields there is a πΆ β π such that πΆ β©(π2 βͺ{π2 }) βͺ = β . We can suppose that for each πΆ β π, πΆ β π1 β(π1 βͺ π 1 ) β {π1 × {π} × {1} : π β π}. In particular, π {π0 , π2 } β© ππππ πΆ = β for β© each πΆ β π. Hence, πππ πΆ = πππ πΆ and our assumption becomes {πππ πΆ : πΆ β π} = β . Note that for each π β π and πΆ β π, πππ πΆ β© ((π1 + 1) × {π} × {1}) is compact. So, for each π β π, there is πΆπ β π such β© that πππ πΆπ β© ((π1 + 1) × {π} × {1}) = β and πΆπ β πΆπ+1 . Thus, {πππ πΆπ : π β π} =β©β . For each π β π, let π₯π β πΆβ©π ; {π₯π : π β π} is an inο¬nite set as {πππ πΆπ : π β π} = β . Also, {πππ {π₯π : π β₯ π} : π β π} = β . There is an πΌ β π1 such that {π₯π : π β π} β (πΌ + 1) × (π + 1) × {1} = π. As π is compact, πππ {π₯π : π β₯ π} β π for each π β π. This shows that β© {πππ {π₯π : π β₯ π} : π β π} = β β , a contradiction. β‘ Property B. It is easy to verify that the space π is Urysohn. Consider the open ο¬lter base β± = {(πΌ, π1 ) × (π, π)} × {1} β© :πΌ β π1 , π β π} on π. The open ο¬lter β± has the property that {πππ πΉ : πΉ β β±} = β . This shows that π is not H-closed. Combining these two results with that of Property A, we have that π is a Urysohn, non-H-closed space with the property that for every chain π of β© nonempty sets in π, {πππ πΆ : πΆ β π} = β β . Our next goal is to show that the Urysohn, non-H-closed space π has the property that for every inο¬nite subset π΄ of π, there is a point π β π such that β£π΄ β© πππ π β£ = β£π΄β£ for all π β π β π (π); we say that π is a πβcomplete accumulation point of π΄. In the following result we will use Alexandroο¬ and Urysohnβs result [1] that every inο¬nite subset of an H-closed space has a π-complete accumulation point. 306 J. PORTER Property C. Every inο¬nite set π΄ of π has a π-complete accumulation point. Proof: Let π΄ be an inο¬nite subset of π. If β£π΄β© (π0 βͺ {π0 })β£ = β£π΄β£, then, as π0 βͺ {π0 } is H-closed, π΄ has a π-complete accumulation point in π0 βͺ {π0 }. But a π-complete accumulation point in π0 βͺ {π0 } is also a π-complete accumulation point in π. A similar result holds for the H-closed subspace π2 βͺ {π2 }. We can suppose that π΄ βͺ β π1 β(π1 βͺ π 1 ) and, as noted in Property A, π1 β(π1 βͺ π 1 ) β {π1 × {π} × {1} : π β π}. If π΄ is a countable set, there is some πΌ < π1 such that π΄ is contained in the compact subspace (πΌ + 1) × (π + 1) × {1}. Thus, π΄ has a complete accumulation point in π. Next, let π΄ be an inο¬nite subset such that β£π΄β£ = π1 . There is some π β π, such that β£π΄ β© ((π1 + 1) × {π} × {1})β£ = π1 . As (π1 + 1) × {π} × {1} is compact for each π β π, π΄ has a complete accumulation point in π. β‘ For an inο¬nite cardinal π and a space π , a function π : π <π β π is called a π -net (see [3]); the π -net is denoted as β¨π₯πΉ : πΉ β π <π β© where π (πΉ ) = π₯πΉ . A point π β π is a π-cluster point of β¨π₯πΉ : πΉ β π <π β© if for π β π β π (π ) and πΉ β π <π , there is πΊ β π <π such that πΉ β πΊ and π₯πΊ β πππ π . The concept of π -nets ο¬ts somewhere between nets and ο¬lters. Hodel [3] proves that a Hausdorο¬ space π is H-closed if and only if every π -net has a π-cluster point in π. A sequence β¨π₯πΌ : πΌ β π β© in a space π has a π-cluster point π β π if for each πΌ β π and π β π β π (π ), there is some π½ β π such that πΌ < π½ and π₯π½ β πππ π . Let π(πΌ) = {π½ β π : πΌ < π½}. Hodel [3] shows that in an H-closed space, every π -sequence has a π-cluster point and asks if a Hausdorο¬ space π in which every π -sequence has a π-cluster point in π for each π is necessarily H-closed. We answer this question by showing that our space π has the property that every π -sequence has a cluster point. First, notice that a π sequence β¨π₯πΌ : πΌ β© β π β© in a space π has a π-cluster point π β π if and only if π β {πππ π(πΌ) : πΌ β π }. Property D. The non-H-closed space π has the property that every π -sequence has a π-cluster point in π. Let β¨π₯πΌ : πΌ β π β© be a π -sequence in π for some inο¬nite cardinal π . The family of sets {π(πΌ) : πΌ β π } of π is a chain β© of nonempty sets. By Property A, there is some point π β {πππ π(πΌ) : πΌ β π }. By the above comment, π is a π-cluster point in π. ALMOST H-CLOSED 307 Comment: By Property A, for every chain π of nonempty sets β© in π, {πππ πΆ : πΆ β π} = β β . If π β² is a chain of nonempty regular closed sets in π, then for each πΆ β π β² , πππ πππ‘π πΆ = πππ πππ‘π πΆ = πΆ. Thus,β©{πππ‘π πΆ : πΆ β π}β©is a chain of nonempty open sets. It follows that {πΆ : πΆ β π} = πππ πππ‘π πΆ : πΆ β π} = β β . That is, the space π has the chain property of the space πΉ in section 2. However, the space πΉ is Tychonoο¬, whereas the space π is only Urysohn. If π is a regular space satisfying the property that for every chain π of nonempty sets in π, β©{πππ πΆ : πΆ β π} = β β , then π is compact because in a regular space, πππ π΄ = πππ π΄ for each subset π΄ of π. For this reason, the space πΉ is also included. Note: Another application of the chain characterization of compactness is in [7] where it is shown that an H-closed space in which every closed set is the π-closure of another set is compact. Many of the ideas in this paper are connected by this very nice result from [3]. We are indebted to the referee for many helpful comments and for suggesting that this result be added. Theorem 4 ([3]). The following are equivalent for any space π and inο¬nite cardinal π : (1) every π-sequence {π₯πΌ : πΌ < π} with π β€ π has a π-cluster point; (2) if {πΉπΌ : πΌ < π} is a decreasing sequence of nonempty subsets of π with π β€ π , then β©πππ (πΉπΌ ) β= β ; (3) if π is a chain of nonempty subsets of π with β£πβ£ β€ π , then β©πππ (πΆ) β= β ; and (4) if π΄ is an inο¬nite subset of π with β£π΄β£ β€ π and β£π΄β£ regular, then π΄ has a π-complete accumulation point. References [1] Paul Alexandroο¬ and Paul Urysohn, Zur theorie der topologischen rauΜme, Math. Ann. 92 (1924), 258β262. [2] Leonard Gillman and Meyer Jerison, Rings of Continuous Functions. The University Series in Higher Mathematics. Princeton, N.J.-Toronto-LondonNew York: D. Van Nostrand Co., Inc., 1960 [3] R. E. Hodel, A theory of convergence and cluster points based on π -nets, Topology Proc. 35 (2010), 291β330. 308 J. PORTER [4] Miroslav KateΜtov, UΜber H-abgeschlossene und bikompakte raΜume, CΜasopis PeΜst. Mat. Fys. 69 (1940), 36β49. [5] G. A. Kirtadze, Diο¬erent types of completeness of topological spaces (Russian), Mat. Sb. (N.S.) 50 (92) (1960), 67β90. [6] A. V. Osipov, Weakly π»-closed spaces, Proc. Steklov Inst. Math. 2004, Topol. Math. Control Theory Diο¬er. Equ. Approx. Theory, suppl. 1, S15β S17. [7] Jack Porter and Mohan Tikoo, On KateΜtov spaces, Canad. Math. Bull. 32 (1989), no. 4, 425β433. [8] Jack R. Porter and R. Grant Woods, Extensions and Absolutes of Hausdorο¬ Spaces. New York: Springer-Verlag, 1988. [9] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375β481. [10] N. V. VelicΜko, H-closed topological spaces, Amer. Math. Soc. Transl. 70(2) (1968), 103β118. Department of Mathematics; University of Kansas; Lawrence, KS 66045 E-mail address: [email protected]