Countably compact and sequentially compact spaces.

*(English)*Zbl 0562.54031
Handbook of set-theoretic topology, 569-602 (1984).

[For the entire collection see Zbl 0546.00022.]

This is an exposition of countable compactness, sequential compactness, and other related properties such as total countable compactness and \(\omega\)-boundedness. There are many examples given throughout which relate these concepts. The two major questions studied are: when do products have these properties; and what must be added to the weaker properties to obtain the stronger ones? Included in one of the two sections on products is a characterization of those Tychonoff spaces whose product with every countably compact Tychonoff space is countably compact. Also the concepts of r-limit and r-compactness are studied, where r is a free ultrafilter on \(\omega\). The concept of r-limit is used to characterize when a product is countably compact, and r-compactness is used to characterize \(\omega\)-boundedness and powers of a space being countably compact. A number of properties are shown to be sufficient to give sequential compactness or compactness when combined with countable compactness. There is also a proof that (under the right set-theoretic assumption) every countably compact perfect \(T_ 3\)-space is compact (and reference is made to a model in which this is not true). The last section contains useful historical notes on the material in each of the preceding sections.

This is an exposition of countable compactness, sequential compactness, and other related properties such as total countable compactness and \(\omega\)-boundedness. There are many examples given throughout which relate these concepts. The two major questions studied are: when do products have these properties; and what must be added to the weaker properties to obtain the stronger ones? Included in one of the two sections on products is a characterization of those Tychonoff spaces whose product with every countably compact Tychonoff space is countably compact. Also the concepts of r-limit and r-compactness are studied, where r is a free ultrafilter on \(\omega\). The concept of r-limit is used to characterize when a product is countably compact, and r-compactness is used to characterize \(\omega\)-boundedness and powers of a space being countably compact. A number of properties are shown to be sufficient to give sequential compactness or compactness when combined with countable compactness. There is also a proof that (under the right set-theoretic assumption) every countably compact perfect \(T_ 3\)-space is compact (and reference is made to a model in which this is not true). The last section contains useful historical notes on the material in each of the preceding sections.

Reviewer: R.A.McCoy

##### MSC:

54D30 | Compactness |

54D45 | Local compactness, \(\sigma\)-compactness |

54B10 | Product spaces in general topology |