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Initially $$\kappa$$-compact and related spaces. (English) Zbl 0588.54025
Handbook of set-theoretic topology, 603-632 (1984).
[For the entire collection see Zbl 0546.00022.]
A topological space X is called [$$\theta$$,$$\kappa$$ ]-compact if every open cover of X of cardinality $$\leq \kappa$$ has a subcover of cardinality $$<\theta$$. If $$\theta =\omega$$ then X is called initially $$\kappa$$- compact. The concept of [$$\theta$$,$$\kappa$$ ]-compactness dates back to P. Alexandroff and P. Urysohn and it has been studied intensively since J. Novak’s discovery in the early 50’s that the product of two countably compact $$(=$$ initially $$\omega$$-compact) spaces can fail to be countably compact.
The author focuses on the question, when is a product of initially $$\kappa$$-compact spaces initially $$\kappa$$-compact. There are some properties stronger than initial $$\kappa$$-compactness which force certain product spaces to be initially $$\kappa$$-compact. Among them the author discusses the following concepts: $$\kappa$$-boundedness, $$\kappa$$- totalness, total initial $$\kappa$$-compactness. It is proved that if X is a totally initially $$\kappa$$-compact space and Y is an initially $$\kappa$$-compact space, then $$X\times Y$$ is initially $$\kappa$$-compact; and that the product of $$\kappa$$-total spaces is $$\kappa$$-total (and hence initially $$\kappa$$-compact). Special attention is also given to constructing some initially $$\kappa$$-compact subspaces of the Čech- Stone compactification of a discrete space whose product fails to be $$\kappa$$-compact. The generalized continuum hypothesis allows one to get such subspaces for every regular uncountable cardinal $$\kappa$$ and, if Martin’s axiom holds, one can construct two subspaces of $$\beta$$ $$\omega$$ that are initially $$\theta$$-compact for every $$\theta <2^{\omega}$$ but their product fails to be countably compact.
There still remain a lot of unsolved problems in the theory of initially $$\kappa$$-compact spaces. We list one of them: is it true that the only cardinal numbers for which initial $$\kappa$$-compactness is productive are singular, strong limit cardinals?
Reviewer: A.Szymański

##### MSC:
 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D30 Compactness 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)