## 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.)

Zbl 0479.54001
Full Text: