Some classes of countably compact spaces
1964; Springer Nature; Volume: 14; Issue: 1 Linguagem: Inglês
10.21136/cmj.1964.100598
ISSN1572-9141
Autores Tópico(s)Advanced Banach Space Theory
ResumoThe present paper investigates the relations between some classes of count ably compact spaces introduced by Z. FROLIK [1], [2].The spaces considered here are always completely regular T^-spaces; N denotes the set of positive integers.Recently, Z. FROLIK introduced the classes ф, ^p and ê which are characterized by the following: A space X belongs to ф (or ^) if and only if for every pseudocompact (or countably compact) space Y the topological product X X У is pseudocompact (or countably compact); a space X belongs to ^p if and only if every closed subspace of X belongs to $.Moreover, he gave necessary and sufficient conditions for a space to belong to one of these classes (see § 1 below).Let ^c be the subclass of ^ consisting of countably compact spaces.It is easy to see that Ф -ф^ is not empty: for instance, X = [1, со] x [1, O] -{(oe, Q)} belongs to "Ф -Фс» where со and Q are the least ordinal numbers of the second and the third classes respectively.In this paper, we shall give new characterizations of ^p in § 2, and consider, in § 3, the relations between the classes ф^, ^p and ё, and show that, the three classes ^ -Фс9 "Фс ^ S and ф^ n (ß -фр) are not empty.Equivalently, i) there is a count ably compact space X such that X x 7is countably compact for every countably compact space Y, but X x Z is not pseudocompact for some pseudocompact space Z, ii) there is a countably compact space X such that X x Yis pseudocompact for every pseudocompact space Y but X x Z is not countably compact for some countably compact space Z, and iii) there is a countably compact space X such that X x У is countably compact (or pseudocompact) for every countably compact (or pseudocompact) space У but X contains some closed subspace Ä having the property that Ä X В is not pseudocompact for some pseudocompact space B, 1. Preliminary.In this section, for convenience, we.shall state Frolik's theorems, and transfer the form 1.3 to the forms 1.5 and 1.6.1.1 [1,3.6].$ Э X if and only if X satisfies the following condition: If 2i is an infinite disjoint family of non-void open subsets of X, then there exists a disjoint sequence
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