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Initially \(\kappa \)-compact spaces for large \(\kappa \). (English) Zbl 0976.54022

Let \(X=\langle X,\tau_X\rangle \) be a \(T_2\) space. The closed pseudocharacter \(\psi_c(X)\) of \(X\) is defined as \(\sup \{\psi_c(p,X)\:p\in X\}+w\), where \(\psi_c(p,X)=\min \{|V|\:V\subseteq \tau_X\), \(p\in \bigcap V\), \(\bigcap \{\overline {V}\:p\in V\}=\{p\}\}\).
The author gives a new bound for \(\psi_c(X)\:\psi_c(X)\leq 2^{d(X)}\) and for a Uryson space \(X\:\psi_c \leq 2^{s(X)}\), where \(d(X)\) and \(s(X)\) are common cardinals for topological spaces [see, e.g., I. Juhász, Cardinal functions in topology – ten years later, Mathematical Centre Tracts 123 (1980; Zbl 0479.54001)].
If every open cover of the space \(X\) of size \(k\) (\(k\geq w\) is a cardinal) has a finite subcover, then \(X\) is said to be initially \(k\)-compact. If for every \(A\subseteq X\) with \(|A|\leq k\) there is \(Y\subseteq X\), \(Y\) compact such that \(A\subseteq Y\), then \(X\) is called \(k\)-bounded. Using the first bound for \(\psi_c(X)\) it is proved in this paper that each \(T_2\) space \(X\) initially \(k\)-compact is \(\lambda \)-bounded for every cardinal \(\lambda \) with \(2^{\lambda}\leq k\).
This result improves the known statement for \(T_3\) spaces and it is used for the proposition on initially \(k\)-compact \(T_2\) spaces being compact for special cardinal \(k\).

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Citations:

Zbl 0479.54001
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