@ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 1, 2012 pp. 27-31 Hausdorff closed extensions of pre-uniform spaces Adalberto Garćıa-Máynez and Rubén Mancio-Toledo Abstract The family of densely finite open covers of a Hausdorff space X deter- mines a completable pre-uniformity on X and the canonical completion ̂X is Hausdorff closed. We compare ̂X with the Katetov extension kX of X and give sufficient conditions for the non-equivalence of kX and ̂X. 2010 MSC: 54A20, 54E15 Keywords: Hausdorff closed space, densely finite-cover, Katetov extension, round filter 1. Preliminary results For the sake of convenience to the reader, we recall some definitions. They also appear in [2]. A filter T in a pre-uniform space (X, U) is U-Cauchy if for every cover α ∈ U, we have T ∩ α �= ∅. A U-Cauchy filter T in a pre-uniform space is U-round if for every F0 ∈ T , there exists a cover α ∈ U such that S∗T (T , α) ⊂ F0, where : S∗T (T , α) = ⋃ {A ∈ α|A ∩ F �= ∅ for every F ∈ T } . U-round filters T in a Hausdorff pre-uniform space (X, U) satisfy the follow- ing conditions: (See [2, Theorem 3.8.4 and 3.8.5]) . 1) For every p ∈ X, T adheres to p if and only if T converges to p. 2) Every neighborhood filter is U-round 28 A. Garćıa-Máynez and R. Mancio-Toledo As a consequence of 1), in Hausdorff pre-uniform spaces, a U-round filter T is either non-adherent or converges to a unique point. An ultrafilter of open sets in a topological space (X, τ) is a non-empty sub- family G of τ − {∅} satisfying : 1) If G1, G2 ∈ G, also G1 ∩ G2 ∈ G; 2) If G ∈ G and G ⊆ H, where H ∈ τ, then H ∈ G; 3) If G0 ∈ τ and G0 ∩ G �= ∅ for every G ∈ G, then G0 ∈ G. Likewise U-round filters, an ultrafilter of open sets in a Hausdorff space X is either non-adherent or converges to a unique point. Hausdorff closed spaces are characterized by the property : ([1, p.283]) *) Every ultrafilter of open sets is convergent. An open cover α of a topological space X is densely finite if there exists a finite subfamily {A1, A2, . . . , An} ⊆ α such that X = A−1 ∪ A − 2 ∪ · · · ∪ A−n . The family U of densely finite covers of a Hausdorff space (X, τ) constitutes a compatible pre-uniform basis which satisfies the condition : **) Every U-Cauchy filter contains a U-round filter. By [5], (X, U) has a canonical completion (X̂, Û) and the topology τ ̂U is Hausdorff closed. X̂ consists of all the U-round filters and Û consists of all the extension covers α̂ (α ∈ U), where α̂ = {  | A ∈ α } and  = { ξ ∈ X̂ | A ∈ ξ } . The canonical embedding h: X → X̂ assigns to each p ∈ X, its neighborhood filter μp. Theorem 2.6 in [3] establishes that a non-adherent filter T in (X, U) is U- round if and only if T has as a basis an ultrafilter of open sets. Besides the completion (X̂, Û), (X, τ) has its Katetov extension kX, where: kX = X ∪ {G | G is a non-adherent ultrafilter of open sets} If p ∈ X, a neighborhood basis of p is the filter μp of τ-neighborhoods of p. If G ∈ kX − X, a neighborhood basis of G consists of all the sets {G} ∪ G, where G ∈ G. The resulting topology of kX turns out to be Hausdorff closed and kX − X is a closed discrete subspace without interior points, and hence X is open and dense in kX. We wonder what is the relation between kX and X̂. We recall first some definitions : A subset A of a topological space X is C-bounded (or relatively pseudocom- pact) if for every continuous function ϕ: X → R, ϕ(A) is bounded. A ⊆ X is C-discrete (with respect to X) if for each a ∈ A, there exists an open set Ua such that a ∈ Ua and the family {Ua | a ∈ A} is discrete (with respect to X). The following equivalence is well known (see, for instance [4] : 4.73.3). A subset A of a Tychonoff space X is C-bounded if and only if every C- discrete subset of X contained in A is finite. Hausdorff closed extensions of pre-uniform spaces 29 An open set U in a topological space X is wide if there exist two open sets W1, W2 such that : 1) W1 ∪ W2 ⊆ U; 2) W1 ∩ W2 = ∅; 3) W −1 and W − 2 are non-compact. For instance, every non-empty open set U in a nowhere locally compact regular space is wide. We also have : Lemma 1.1. Every open set U in a Tychonoff space which is not C-bounded, is wide. Proof. By hypothesis, there exists an infinite discrete family of open sets W1, W2, . . . , where W − i ⊆ U for every i. If S = ⋃∞ i=1 W2i−1 and T = ⋃∞ i=1 W2i, we have S− ∪ T − ⊆ U, S ∩ T = ∅ and none of the sets S−, T − is compact. � 2. Main result We give a sufficient condition on a Tychonoff space X which insures that the extensions X̂ and kX are non-equivalent. Theorem 2.1. Let X be a non-compact Tychonoff space where every open set with non-compact closure is wide. Then X̂ − h(X) is dense in itself, where h: (X, U) → (X̂, Û) is the canonical embedding of X into X̂. Proof. Let us take any element ξ ∈ X̂ − X (we identify each point p ∈ X with its neighborhood filter). Let U be an open set in X such that ξ ∈ Û. Therefore, U ∈ ξ. Since the round filter ξ is non-adherent, U− cannot be compact. By hypothesis, U is wide. Let S, T be open sets such that S ∪T ⊆ U, S ∩ T = ∅ and S−, T − are both non-compact. By [1, p. 283], S− and T − cannot be Hausdorff closed. Hence there exist non-adherent ultrafilters of open sets μ1, μ2 in S −, T −, respectively. Hence the restrictions μ1 | S and μ2 | T are non-adherent filterbases consisting of open sets in X. Take ultrafilters of open sets ξ1, ξ2 in X containing μ1 | S and μ2 | T , respectively. Clearly ξ1 and ξ2 are also non-adherent and U belongs to both of them. Therefore, at least one of the round filters ξ+1 , ξ + 2 is different from ξ. Therefore, Û ∩ (X̂ − X) consists of more than one element and X̂ − X is dense in itself. � Corollary 2.2. Every normal Hausdorff metacompact space X satisfies the condition in the theorem. Proof. Let U ⊆ X be an open set whose closure is non compact. By [4, 4.74.5], the subspace U− cannot be pseudocompact and hence U cannot be C-bounded. � 30 A. Garćıa-Máynez and R. Mancio-Toledo Corollary 2.3. t If X is paracompact and T2, then X̂ − X is dense in itself, and hence the extensions kX and X̂ are non-equivalent (unless X is compact). Lemma 2.4. Let U be an open set in a regular Hausdorff space X and let ξ ∈ Û ∩ (X̂ − X). Then U is wide if and only if (Û − {ξ}) ∩ (X̂ − X) �= ∅. Hence, if U is not wide, we have {ξ} = Û ∩ (X̂ − X). Proof. Reason as in Theorem 2.1. � Example 2.5. Let X be the space of countable ordinals with the order topo- logy. Then every uncountable open set in X is wide and hence X̂ − X is dense in itself. Proof. Let D be the set of non-limit ordinals in X. Then D is open, discrete and dense in X. If U ⊆ X is an uncountable open set in X, then U ∩ D is also uncountable (because otherwise U− = (U ∩ D)− would be compact and hence U would be countable). Clearly, U ∩D is the union of two uncountable disjoint subsets. Hence, U is wide. � Example 2.6. The half disk X = { (p, q) ∈ R2 | p2 + q2 ≤ 1, q > 0 } has a non- compact Hausdorff closed extension Z whose remainder Z − X is closed and discrete. However Z is not equivalent to X̂ neither to kX. Proof. Let Z = X ∪ {(z, 0) | − 1 ≤ z ≤ 1}. For each (z, 0) ∈ Z − X, define μz be the set of unions of {(z, 0)} with upper half open disks in R2 centered at (z, 0) and intersected with X. If z ∈ X, μz consists of all open disks in R2 centered at z and intersected with X. We can now topologize Z with the help of these filter bases μz and convert it into a Hausdorff, non-regular, extension of X. To see that Z is Hausdorff closed, we consider a cover of Z consisting of elements of the filterbases μz. Since { (p, q) ∈ R2 | p2 + q2 ≤ 1, q ≥ 0 } is compact in the usual topology of R2, we could get a finite subcover for this space if we adjoin to the elements of μz (z ∈ Z − X) their radii in the X-axis. Therefore, the original cover has a finite subfamily which covers X (recall X is dense in Z). This argument proves that every open cover of Z is densely finite, and hence Z is Hausdorff closed (see [1]). Clearly the remainder Z − X is closed and discrete. For each point z ∈ Z − X, we can find an infinite family of ultrafilters of open sets in X which have z as a convergence point. This remark proves that Z is not equivalent to X̂ neither to kX, because in these extensions, every point of the remainder is the convergence point of a unique ultrafilter of open sets in X. � Hausdorff closed extensions of pre-uniform spaces 31 References [1] R. Engelking, General Topology, Polish Scientific Publishers (Warszawa, 1975). [2] A. Garćıa-Máynez and Rubén S. Mancio, Completions of pre-uniform spaces, Appl. Gen. Topol. 8, no. 2 (2007), 213–221. [3] A. Garćıa-Máynez and Rubén S. Mancio, Extending maps between pre-uniform spaces, Appl. Gen. Topol. 13, no. 1 (2012), 21–25. [4] A. Garćıa-Máynez and A. Tamariz, Topoloǵıa General, Porrúa (México, 1988). [5] Rubén S. Mancio, Los espacios pre-uniformes y sus completaciones, Ph. D. Thesis, Universidad Nacional Autónoma de México, 2007. (Received January 2010 – Accepted January 2011) A. Garćıa-Máynez (agmaynez@matem.unam.mx) Instituto de Matemáticas, Universidad Nacional Autónoma de México, Área de la Investigación Cient́ıfica, Circuito Exterior, Ciudad Universitaria, 04510 México, D.F. México Rubén Mancio-Toledo (rmancio@esfm.ipn.mx) Escuela Superior de F́ısica y Matemáticas, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Col. Lindavista, 07738 México, D.F. Hausdorff closed extensions of pre-uniform[6pt] spaces. By A. García-Máynez and R. Mancio-Toledo