Á. Tamariz-Mascarúa

On Cα-compact subsets

Abstract For an infinite cardinal α, we say that a subset B of a space X is Cα-compact in X if for every continuous function f : X → Rα, [B] is a compact subset of Rα. This concept slightly generalizes the notion of α-pseudocompactness introduced by J.F. Kennison: a space X is α-pseudocompact ifX is Cα-compact in itself. If α = ω, then we say C-compact instead of Cω-compact and ω-pseudocompactness agrees with pseudocompactness. We generalize Tamanos theorem on the pseudocompactness of a product of two spaces as follows: let A ⊆ X and B ⊆ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A × B is Cα-compact in X × Y; (2) A and B are Cα-compact in X and Y, respectively, and the projection map π : X × Y → X is a zα-map with respect to A × B and A; and (3) A and B are Cω-compact in X and Y, respectively, and the projection map π : X × Y → X is a strongly zα-map with respect to A × B and A (the zα-maps are the strongly zα-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of Cα-compactness of a C-compact subset B of a space X is defined by: ϱ(B,X) = ∞ if B is compact, and if B is not compact, then ϱ(B,X) = supα: B is Cα-compact in X. We estimate the degree of pseudocompactness of locally compact pseudocompact spaces, topological products and Σ-products. We also establish the relation between the pseudocompact degree and some other cardinal functions. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space (X,U), then A is Cα-compact in X , where ( X , U ), is the completion of (X,U) iff f(A) is a compact subset of Rα from every uniformly continuous function from X into Rα; we characterize the Cα-compact subsets of topological groups; and we also prove that if Gi: i I is a set of topological groups 15 and Ai is a Cα-compact subset of Gα for all i I, then ΠiI Ai is a Cα-compact subset of ΠiI Gi.

p-pseudocompactness and related topics in topological spaces

Abstract We prove some basic properties of p-bounded subsets ( p∈ω ∗ ) in terms of z-ultrafilters and families of continuous functions. We analyze the relations between p-pseudocompactness with other pseudocompact like-properties as p-compactness and α-pseudocompactness where α is a cardinal number. We give an example of a sequentially compact ultrapseudocompact α-pseudocompact space which is not ultracompact, and we also give an example of an ultrapseudocompact totally countably compact α-pseudocompact space which is not q-compact for any q∈ω ∗ , answering affirmatively to a question posed by S. Garcia-Ferreira and Kocinac (1996). We show the distribution law clγ(X×Y)(A×B)=clγXA×clγYB, where γZ denotes the Dieudonne completion of Z, for p-bounded subsets and we generalize the classical Glisckberg Theorem on pseudocompactness in the realm of p-boundedness. These results are applied to study the degree of pseudocompactness in the product of p-bounded subsets.

p-Fréchet-Urysohn property of function spaces

In this paper, we study the p-FrCchet-Urysohn property of function spaces, for p E /3(w)\w. We prove that C,(X) is p-FrCchet-Urysohn if and only if X has (r,), where (7,) is the natural p-version of property (y) (this is a generalization of a result due to Gerlits and Nagy). We note the following implications: X is second countable *X has (7,) for some p EP(o)\o -X n is LindelGf for all 1 Q n \w such that C,(R) is p-FrCchet-Urysohn; if p is semiselective, then every subset X of R satisfying (r,,) has measure zero and if p is selective, then X is a strong measure zero set; and we can find p E /3(o>\o such that C,(R) is p-FrCchet-Urysohn and is not strongly p-FrCchet-Urysohn. Finally, we prove that [w” does not have (-y,) whenever p is a P-point of P(w)\w.

Extensions of functions in Mrówka-Isbell spaces

Abstract For an almost disjoint family (a.d.f.) ∑ of subsets of ω, let Ψ(∑) be the Mrowka-Isbell space on ∑. In this article we will analyze the following problem: given an a.d.f. ∑ and a function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) is it possible to extend φ continuously to a big enough subspace ∑ ∪ N of Ψ(∑) for which lΨ(∑)N ⊃ ∑? Such an extension is called essential. We will prove that: 1. (i) for every a.d.f. ∑ of cardinality 2ℵ0 we can find a function φ: ∑ → {0, 1} without essential extensions; 2. (ii) for every m.a.d. family ∑ there exists a function φ: ∑ → R that has no essential extension; and 3. (iii) there exists a Mrowka-Isbell space Ψ(∑) of cardinality ℵ1 such that every function φ: ϵ → R with at least two different uncountable fibers, has no full extension. On the other hand, under Martins Axiom every function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) has an essential extension if ¦∑¦ ℵ 0 . Finally, we analyze these questions under CH and by adding new Cohen reals to a ground model M showing that the existence of an uncountable a.d.f. ∑ for which every onto function φ: ∑ → {0, 1} with infinite fibers has no essential extensions is consistent with ZFC.

Weakly Pseudocompact Spaces

A well known result established by Hewitt (Trans Amer Math Soc 64:45–99 1948, [16]) states that a space X is pseudocompact if and only if X is \(G_\delta \)-dense in \(\beta X\). In Garcia-Ferreira and Garcia-Maynez (Houston J Math 20(1):145–159, 1994, [12]), S. Garcia-Ferreira and A. Garcia-Maynez introduced the following concept: a topological space is weakly pseudocompact if it is \(G_\delta \)-dense in one of its compactifications. Thus, every pseudocompact space is weakly pseudocompact.

The partially pre-ordered set of compactifications of Cp(X, Y)

Abstract In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R). We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen: (a) Y is not locally compact, (b) X has only one non isolated point and Y is not compact. Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties: (i) X has a non-isolated point with countable character, (ii) X is not pseudocompact, (iii) X is infinite, pseudocompact and Cp(X) is normal, (iv) X is an infinite generalized ordered space. Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC,

Countable product of function spaces having p-Frechet-Urysohn like properties

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