Spaces whose Pseudocompact Subspaces are Closed Subsets

Abstract

Every first countable pseudocompact Tychonoff space X has the property that every pseudocompact subspace of X is a closed subset of X (denoted herein by "FCC"). We study the property FCC and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. Characterization, embedding and product theorems are obtained, and some examples are given which provide results such as the following. There exists a separable Moore space which has no regular, FCC extension space. There exists a compact Hausdorff Fréchet space which is not FCC. There exists a compact Hausdorff Fréchet space X such that X, but not X 2 , is FCC.

The equivalence of two definitions of sequential pseudocompactness

“…In this note we show their equivalence. Putting together the results from [1] and [2] shows that the class of sequentially pseudocompact Tychonoff topological spaces is closed under (possibly infinite) products and contains significant classes of pseudocompact spaces.…”

On old and new classes of feebly compact spaces

“…On the other hand, the space Y is not totally countably pracompact. For this purpose it suffices to show that for any point x = (x α ) ∈ Y a set {x} (everywhere in this example we by S we mean closure in Y of its subset S) is not compact , because {x} is closure (in Y ) of any subsequence of a constant sequence {x n }, where x n = x for each n. By [11,Proposition 2 Since sequential feebly compactness is preserved by extensions, the next proposition strengthen a bit Theorem 4.1 of [10].…”

“…But already the Cantor cube D c is not sequentially compact (see [11], the paragraph after Example 3.10.38). On the other hand some compact-like spaces are also preserved by products, see [ [10] proved that a product of a family of sequentially feebly compact spaces is again sequentially feebly compact, and in Theorem 4.3 that every product of feebly compact spaces, all but one of which are sequentially feebly compact, is feebly compact.…”

“…Put B = B n . By Theorem 4.1 of [10], the space X ′ = {X α : α ∈ B} is sequentially feebly compact. Since {π B (U n )} is a sequence of its non-empty open subsets, there exist a point x ′ ∈ X ′ and an infinite set I ⊂ N such that for each neighborhood U ′ of the point…”

Maximalidad de propiedades que generalizan la compacidad

“…A este punto p lo llamaremos P-punto crítico de la sucesión de conjuntos abiertos hU n i. Un espacio con la propiedad (P 0 ) es llamado secuencialmente tenuemente compacto en [17].…”