summary:A non-connected, Hausdorff space with a countable network has a connected Hausdorff-subtopology iff the space is not-H-closed. This result answers two questions posed by Tkačenko, Tkachuk, Uspenskij, and Wilson [Comment. Math. Univ. Carolinae 37 (1996), 825--841]. A non-H-closed, Hausdorff space with countable $\pi $-weight and no connected, Hausdorff subtopology is provided

Porter, Jack

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Commentationes Mathematicae Universitatis CarolinaeJack R. PorterConnected Hausdorff subtopologiesCommentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 1, 195--201Persistent URL: http://dml.cz/dmlcz/119234Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 42,1 (2001)195–201 195Connected Hausdorff subtopologiesJack PorterAbstract. A non-connected, Hausdorff space with a countable network has a connectedHausdorff-subtopology iff the space is not-H-closed. This result answers two questionsposed by Tkačenko, Tkachuk, Uspenskij, and Wilson [Comment. Math. Univ. Carolinae37 (1996), 825–841]. A non-H-closed, Hausdorff space with countable π-weight and noconnected, Hausdorff subtopology is provided.Keywords: connected, H-closed, extensions, condensationsClassification: 54C10, 54D05, 54D35IntroductionLet X be a space. A topology σ onX is a subtopology of τ(X) if σ ⊆ τ(X). Theaim of this paper is to determine when a space has a connected, Hausdorff sub-topology. Tkačenko, Tkachuk, Uspenskij, and Wilson [TTUW] have establishedthese two results:(1) A countable infinite Hausdorff space has a connected, Hausdorff subtopo-logy iff it is not H-closed.(2) A nonconnected, T3 space with a countable network has a connected, Haus-dorff subtopology iff it is not compact.In this paper we extend (1) and (2) and completely answer two of the questionsposed in [TTUW] by proving this result:Main Theorem. A nonconnected, Hausdorff space with a countable networkhas a connected Hausdorff subtopology iff it is not H-closed.Examples are provided to show that the hypothesis property of countable net-work in the main theorem cannot be replaced by a countable π-weight or 2ω-network. Vermeer [V] defined a Hausdorff space to be absolute Katětov if ev-ery Hausdorff subtopology has an H-closed subtopology and noted that H-closedspaces are absolute Katětov. He asked if every absolute Katětov is H-closed. Weshow that a countable Hausdorff space is absolute Katětov iff it is H-closed andprovide an example of a non-H-closed space that is absolutely Katětov.We extend the well-known result that a compact Hausdorff space with a count-able network is second countable to this result: if X is an H-closed space witha countable network, then X(s) is second countable. An example of an H-closedspace with a countable network is provided which is not second countable.196 J. PorterFirst some basic definitions (see [PW1]) are provided.A Hausdorff space X is H-closed if whenever Y is a Hausdorff space and Xis a subspace of Y , then X is closed in Y . For a Hausdorff space X , this isequivalent to the property that every open ultrafilter on X converges and to theproperty that for every open cover C of X , there is a finite subset D ⊆ C suchthat X = clX(∪D). A Hausdorff space X is almost H-closed if there is exactlyone free open ultrafilter on X .A space X is feebly compact (see 1.11 in [PW1]) if for every countableopen cover C of X , there is a finite subset D ⊆ C such that X = clX(∪D).A space is not feebly compact iff there is an infinite locally finite family of pairwisedisjoint nonempty open subsets. A Tychonoff space is feebly compact iff it ispseudocompact.Let X be a Hausdorff space and τ(X)(s) be the topology generated by the openbase {intXclX(U) : U ∈ τ}. It is easy to check that τ(X)(s) ⊆ τ(X) and that(X, τ(X)(s)), sometimes denoted asX(s), is also a Hausdorff space. In particular,τ(X(s)) = τ(X)(s). A space X is semiregular if τ(X)(s) = τ(X). The spaceX(s) is semiregular.Let X and Y be two spaces. A function f : X → Y is θ−continuous if foreach p ∈ X and open set U ∈ τ(Y ) such that f(p) ∈ U , there is an open setV ∈ τ(X) such that p ∈ V and f [clXV ] ⊆ clY U .Here are some known results about H-closed spaces and θ−continuous functionsthat will be useful in the sequel.Fact 1. Let X and Y be Hausdorff spaces and f : X → Y be a surjection.(a) If X is H-closed and f is θ−continuous, then Y is also H-closed.(b) If X is connected and f is θ−continuous, then Y is also connected.(c) The space X is H-closed iff X(s) is H-closed.(d) If X is H-closed and σ is a Hausdorff subtopology, then τ(X(s)) ⊆ σ ⊆τ(X).(e) The space X is connected iff X(s) is connected.Note. An easy consequence of Fact 1 is that an H-closed space has a connectedHausdorff subtopology iff it is connected.Let X and Y be sets and f : Y → X be a function. For A ⊆ Y , definef#[A] = {x ∈ X : f←(x) ⊆ A}. Note that for subsets A, B ⊆ Y , f#[Y \A] =X\f [A] and f#[A ∩ B] = f#[A] ∩ f#[B]. The topology on Y generated by{f#[U ] : U ∈ τ(Y )} is called the θ−quotient topology induced by f . Thefunction f is called irreducible if for each nonempty open set U ∈ τ(Y ), thereis some x ∈ X such that f←(x) ⊆ U .Fact 2. Let f : Y → X be onto and compact where Y is a Hausdorff space andX is a set. Let σ be the θ−quotient topology induced by f . Then:(a) (X, σ) is a Hausdorff space,(b) if X is a space and f is closed, then σ ⊆ τ(X),Connected Hausdorff subtopologies 197(c) if f is irreducible, then f is θ−continuous, and(d) if f is irreducible and Y is semiregular, then X is semiregular.Application 3. (a) One obtains an easy proof of this result from [TTUW]:Let Y be a Hausdorff connected extension of a space X . If there is a closed,discrete subset A of X such that |Y \X | ≤ |A|, then X has a connected, Hausdorffsubtopology.[If g : Y \X → A is any one-to-one function, it is straightforward to show thatthe function f : Y → X defined by f(y) = g(y) for y ∈ Y \X and f(x) = x forx ∈ X is a perfect irreducible surjection; apply Fact 2.](b) Let X be a Hausdorff space with a countable π-base B such that for eachB ∈ B, clB is not feebly compact. By a result in [PW2] we know that X hasa connected Hausdorff extension Y such that Y \X is countable. There is aninfinite closed discrete subset as X is not countably compact. Applying (a), Xhas a connected, Hausdorff subtopology.Let X be a Hausdorff space and let ΘX = {U : U is an open ultrafilter on X}.For U ∈ τ(X), let O(U) = {U : U ∈ U}. For U, V ∈ τ(X), it is easy to verify(see [PW1]) that O(∅) = ∅, O(X) = ΘX, O(U ∩V ) = O(U)∩O(V ), O(U ∪V ) =O(U) ∪ O(V ),ΘX\O(U) = O(X\clXU), and O(U) = O(intXclXU). ΘX withthe topology generated by {O(U) : U ∈ τ(X)} is an extremally disconnectedcompact Hausdorff space. The subspace EX = {U ∈ ΘX : U is fixed} is calledthe absolute of X . The function k : EX → X defined by k(U) is the uniqueconvergent point of U is called a covering function. The subspace EX is dense inΘX (in particular, EX is an extremally disconnected Tychonoff space and ΘX =βEX), and the covering function k : EX → X is irreducible, θ−continuous,perfect and onto.A family F of subsets of a space X is a network if for each open set U andp ∈ U , there is an F ∈ F such that p ∈ F ⊆ U . A space X with a countablenetwork F has a coarser second countable Hausdorff topology. This is verifiedby first letting H = {(F, G) ∈ F2 : there are disjoint open sets U, V such thatF ⊆ U and G ⊆ V }. For (F, G) ∈ H, let UFG, VFG be disjoint open sets such thatF ⊆ UFG, G ⊆ VFG. Note that H is countable and so {UFG, VFG : (F, G) ∈ H}generates a second countable topology σ on X such that σ ⊆ τ(X). If p, q ∈ Xare distinct points, there are disjoint open sets U, V ∈ τ(X) such that p ∈ U andq ∈ V . So, there are F, G ∈ F such that p ∈ F ⊆ U and q ∈ G ⊆ V . Now,UFG, VFG are disjoint open sets containing p, q respectively.Thus, the σ is the desired coarser second countable Hausdorff topology.A key lemma from [TTUW] is needed before we can start the proof of the mainresult.Lemma 4 ([TTUW]). A noncompact, separable metrizable space has a separablemetrizable subtopology which is nowhere locally compact.198 J. PorterProof of the Main Theorem. Suppose X is not H-closed and has a countablenetwork for X . As X is Lindelöf and not H-closed, it follows that X is not feeblycompact. Thus, there is a locally finite family {Un : n ∈ ω} of pairwise disjointnonempty open subsets of X . It is easy to verify that {O(Un) : n ∈ ω} is a locallyfinite family of pairwise disjoint nonempty clopen subsets of EX . So, EX is notfeebly compact. As EX is Tychonoff, it follows that EX is not pseudocompactand there is a continuous unbounded real-valued function f0 on EX . There isa countable family {Vn : n ∈ N} of open subsets of X with the property that ifp, q ∈ X and p 6= q, there is some n ∈ N such that p ∈ Vn and q 6∈ clXVn. Now,EX∩O(Vn) is a clopen subset of EX ; let fn be the continuous real-valued functionon EX such that fn[EX∩O(Vn)] = {0} and fn[EX\O(Vn)] = {1}. In particular,for p, q ∈ X and p 6= q, there is some n ∈ N such that fn[k←(p)] = {0} andfn[k←(q)] = {1}. The diagonal function f : EX →∏ω R defined by f(y)(n) =fn(y) for n ∈ ω is continuous (not necessarily one-to-one), f [EX ] is not compactas f0 is unbounded, and f [k←(p)] ∩ f [k←(q)] = ∅ for distinct points p, q ∈ X .By Lemma 4, the space f [EX ] has a separable metrizable subtopology µ whichis nowhere locally compact. By Application 3(b), (f [EX ], µ) has a connected,Hausdorff subtopology σ. Since f [k←(p)] ∩ f [k←(q)] = ∅ for distinct pointsp, q ∈ X , it follows there is function g : f [EX ]→ X such that g ◦ f = k, i.e., thefollowing diagram commutes.EX EXfyyk(f [X ], µ)g−−−−→ Xidf [X]yyidX(f [X ], σ)g−−−−→ Xidf [X]yyidX(f [X ], σ)g−−−−→ (X, ρ)◦◦◦Note that f : EX → (f [X ], σ) is continuous and for p ∈ X , g←(p) = f [k←(p)].Thus, g : (f [X ], σ) → X is a compact function. Clearly, g is onto. If A is aclosed subset of (f [X ], σ), then g[A] = k[f←[A]] is closed in X . So, g is a closedfunction. If ∅ 6= U ∈ σ, then there is a point p ∈ X such that k←(p) ⊆ f←[U ].So, g←(p) = f [k←(p)] ⊆ f [f←[U ]] = U . This shows that g : (f [X ], σ) → X isirreducible.Let ρ be the θ−quotient topology on X induced by g : (f [X ], σ) → X . ByFact 2, (X, ρ) is a Hausdorff space, ρ ⊆ τ(X), and g : (f [X ], σ) → (X, ρ) isθ−continuous. By Fact 1, (X, ρ) is connected. Connected Hausdorff subtopologies 199By using the fact that a countable H-closed space has a dense subset of isolatedpoints [PW1], an easy consequence of the above theorem is the following knownresult which motivated the problem of this manuscript.Corollary ([TTUW]). A countable Hausdorff space has a connected, Hausdorffsubtopology iff it is not H-closed.Notation ([PW1]). Let X be a space and F ,G be filter bases onX . The notationF ≤ G means for each F ∈ F , there is a G ∈ G such that G ⊆ F , and F = Gmeans F ≤ G and G ≤ F .Recall [PV] that a Hausdorff space is Katětov if it has an H-closed subtopology.Corollary. A countable Hausdorff space which is not H-closed has a Hausdorffsubtopology which is not Katětov.Proof: A countable Hausdorff space which is not H-closed has a connected,Hausdorff subtopology and this subtopology has no isolated points. In particular,this subtopology is not Katětov as a countable H-closed space has a dense set ofisolated points. Fact 7. Let X be an almost H-closed space with three pairwise disjoint clopensets. Let σ be a Hausdorff subtopology of X . Then (X, σ) is not connected andeither τ(X)(s) ⊆ σ or (X, σ) is H-closed.Proof: Let σ be Hausdorff subtopology of X . If τ(X(s)) ⊆ σ, then (X, σ) is notconnected as X(s) is not connected by Fact 1(e). Suppose τ(X)(s) 6⊆ σ. Let Ube the free open ultrafilter on X . For each q ∈ X , Fq = {U ∈ τ(X)(s) : q ∈ U}and Gq = {U ∈ σ : q ∈ U} are open filter bases on X . There is some r ∈ Xsuch that Fr 6≤ Gr and there is some V ∈ Fr such that U\V 6= ∅ for all U ∈ Gr.There is some W ∈ Fr such that W = intXclXW ⊆ V . It follows that V ={U\clXW : U ∈ Gr} is a free open filterbase on X . Thus, V ⊆ U and Gr ⊆ U . IfFs 6≤ Gs, then a similar argument shows that Gs ⊆ U . That is, Gr meets Gs. As(X, σ) is Hausdorff, Gr = Gs. Assume that (X, σ) is not H-closed. Then there isa free open filter W on (X, σ). So, W is a free open filter base on X and W ⊆ U .So, Gr meets W , a contradiction. Thus, (X, σ) is H-closed. Of the three pairwisedisjoint clopen sets, at least two do not meet U . So, there is a clopen set C suchthat r /∈ C /∈ U . As Gr ⊆ Fr ∪ U , r /∈ clσC. So, C is closed in (X, σ). AsC ∈ τ(X)(s) and Fs ⊆ Gs for all s ∈ X\{r}, it follows that C ∈ σ. Hence, (X, σ)is not connected. Example. (1) One question is whether the main result is true when the “count-able network” part of the hypothesis is replaced by “countable π-weight”. TheSorgenfrey Line is the usual example of a space with countable π-weight but nocountable network. However, the Sorgenfrey Line has a connected Tychonoff sub-topology (i.e., the real line is a subtopology). Now, βω\{p} where p ∈ βω\ω isalmost H-closed and 0-dimensional. By Fact 7, βω\{p} has no connected Haus-dorff subtopology. Also, βω\{p} has weight 2ω and hence a 2ω-network. So,200 J. Porterthe Main Theorem cannot be improved by replacing the hypothesis of “countablenetwork” by “2ω-network”.(2) Another question is whether the main result is true when the hypothesis of“countable network” is replaced by “cardinality ≤ 2ω”. Here is a counterexample:By repeating the proof of 3.5 in [PW2], there is an almost H-closed extension Xof ω such that |X | = c. By Fact 7, X does not have a connected Hausdorffsubtopology.Comment. Vermeer [V] noted that H-closed spaces are absolute Katětov andasked if there were absolute Katětov spaces which were not H-closed. Vermeer’squestion is re-inforced by the Corollary that the only countable spaces which areabsolute Katětov are the H-closed spaces. However, Fact 7 shows that any almostH-closed space is also absolute Katětov.H-closed plus countable networkNote. A space with a countable network is separable and Lindelöf and has theproperty that every discrete subspace is countable. A compact Hausdorff spacewith a countable network is second countable. A natural question is whether anH-closed space with a countable network is second countable. The answer is yesif the space is also semiregular (i.e., minimal Hausdorff) but an example (afterthe following Fact) shows that an H-closed space with a countable network maynot have a countable π-base.Fact 8. If X is an H-closed space with a countable network, then X(s) is secondcountable.Proof: Let C = {Cn : n ∈ ω} be a countable network for X . Let C2 ={〈Cn, Cm〉 : there are regular open sets Unm and Vnm such that Cn ⊆ Unm,Cm ⊆ Vnm, and Unm ∩ Vnm = ∅}. For each pair 〈Cn, Cm〉 ∈ C2, we se-lect exactly one pair 〈Unm, Vnm〉. Let σ be the topology on X generated by{Unm, Vnm : 〈Cn, Cm〉 ∈ C2}, and note that (X, σ) is a Hausdorff space with acountable base and σ ⊆ τ(X). As X is H-closed, τ(X)(s) ⊆ σ. However, sinceτ(X)(s) is generated by the collection of all regular open sets, it follows thatσ ⊆ τ(X)(s). That is, σ = τ(X)(s). Example. Let X = [0, 1]2, Y = X\([0, 1]×{0}), σ the usual topology on X , andS = {S ⊂ Y ω : there is a bijection f : ω → S converging to (0, 0)}. Note thatS is closed under finite unions. Let τ(X) denote the topology on X generatedby σ ∪ {X\S : S ∈ S}. Note that τ(X)(s) = σ. So, X is H-closed. Let B bea countable base for (X, σ). Then B ∪ {[0, 1n ) × {0} : n ∈ ω\{0}} is a countablenetwork for X . Also, X is not first countable at (0, 0). In fact, there is nocountable π-base at (0, 0). Connected Hausdorff subtopologies 201References[PT] Porter J.R., Tikoo M.L., On Katětov spaces, Trans. Amer. Math. Soc. 289 (4) (1985),59–71.[PV] Porter J.R., Vermeer J., Space with coarser minimal Hausdorff topologies, Canad.Math. Bull. 32.4 (1989), 425–433.[PW1] Porter J.R., Woods R.G., Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, Berlin, 1988.[PW2] Porter J.R., Woods R.G., Subspaces of connected spaces, Topology Appl. 68 (1996),113–131.[V] Vermeer J., Private communication, 1984.[TTUW] Tkačenko M.G., Tkachuk V.V., Uspenskij V.V., Wilson R.G., In quest of weaker con-nected topologies, Comment. Math Univ. Carolinae 37.4 (1996), 825–841.Department of Mathematics, University of Kansas, Lawrence, Kansas 66045 USAE-mail : porter@math.ukans.edu(Received December 17, 1999)

Connected Hausdorff subtopologies

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Connected Hausdorff subtopologies

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Institute of Mathematics AS CR, v. v. i.