##
**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?

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.) |