Papers by A. Tamariz-Mascarúa
ABSTRACT Given p () , we determine when a product of quasi-p-pseudocompact spaces preserves this ... more ABSTRACT Given p () , we determine when a product of quasi-p-pseudocompact spaces preserves this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of () containing . We give examples of spaces X, Y, X s , Ys which are quasi-p-pseudocompact for every p *, but X Y is not pseudocompact, and X s Y s is pseudocompact and it is not quasi-s-pseudocompact for each s *. Besides, we prove that every pseudocompact space X of with X, is quasi-p-pseudocompact for some p *. Finally, we introduce, for each p *, the class P p of all spaces X such that X Y is quasi-p-pseudocompact when so is Y; and we prove: (1) the intersection of classes P p ( p *) coincides with the Frol"ik class; (2) every class P p is closed under arbitrary products; (3) the partial ordered set ( P p p ,) is isomorphic to the set of equivalence classes of free ultrafilters on with the Rudin–Keisler order. A topological characterization of RK-minimal ultrafilters is also given.
Topology and its Applications, 2012
a r t i c l e i n f o a b s t r a c t
Topology and its Applications, 1997
For an infinite cardinal a, we say that a subset B of a space X is Ca-compact in X if for every c... more For an infinite cardinal a, we say that a subset B of a space X is Ca-compact in X if for every continuous function f : X -~ II~ ~, f [B] is a compact subset of II~ ~. This concept slightly generalizes the notion of a-pseudocompacmess introduced by J.F. Kennison: a space X is a-pseudocompact if X is Ca-compact in itself. If a = w, then we say C-compact instead of C~-compact and ~v-pseudocompactness agrees with pseudocompactness. We generalize Tamano's theorem on the pseudocompactness of a product of two spaces as follows: let A C_ X and B C_ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A x B is C~compact in X x Y; (2) A and B are Ca-compact in X and Y, respectively, and the projection map ~r : X x Y ~ X is a za-map with respect to A x B and A; and (3) A and B are Co-compact in X and Y, respectively, and the projection map ~':X x Y -+ X is a strongly z~-map with respect to A x B and A (the z~-maps and the strongly z~-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of C,~-compactuess of a C-compact subset B of a space X is defined by: p(B, X) = cx~ if B is compact, and if B is not compact, then p(B, X) = sup{a: B is Ca-compact in X}. We estimate the degree of pseudocompacmess of locally compact pseudocompact spaces, topological products and E-products. We also establish the relation between the pseudocompact degree and some other cardinal function's. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space ~X,~/), then A is Ca-compact in X, where (X,~) is the completion of (X,/,/) iff f(A) is a compact subset of R ~ from every uniformly continuous function from X into ~'~; we characterize the C~-compact subsets of topological groups; and we also prove that if {Gi: i E I} is a set of topological groups and A~ is a Ca-compact subset of G,~ for all i E I, then I-L~ A~ is a Co-compact subset of 1-[iel Gi. © 1997 Elsevier Science B.V.
Topology and its Applications, 1997
For an almost disjoint family (a.
ABSTRACT Let X be a Hausdorff space and let H be one of the hyperspaces CL(X), K(X), F(X) or Fn(X... more ABSTRACT Let X be a Hausdorff space and let H be one of the hyperspaces CL(X), K(X), F(X) or Fn(X) (n a positive integer) with the Vietoris topol-ogy. We study the following disconnectedness properties for H: extremal disconnectedness, being a F -space, P -space or weak P -space and hereditary disconnectedness. Our main result states: if X is Hausdorff and F ⊂ X is a closed subset such that (a) both F and X − F are totally disconnected, (b) the quotient X/F is hereditarily disconnected, then K(X) is hereditarily disconnected. We also show an example proving that this result cannot be reversed. Given a T 1 space X, let CL(X) be the hyperspace of nonempty closed subsets of X with the Vietoris topology. Let us consider the following hyperspaces K(X) = {A ∈ CL(X) : A is compact}, F(X) = {A ∈ CL(X) : A is finite}, F n (X) = {A ∈ CL(X) : |A| ≤ n} for n a positive integer. The study of the Vietoris topology on hyperspaces was first motivated by Ernest Michael's outstanding paper [M]. Concerning connectedness properties, Michael stated the following:
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Papers by A. Tamariz-Mascarúa