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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.
Reviewer: R.A.McCoy

54D30 Compactness
54D45 Local compactness, \(\sigma\)-compactness
54B10 Product spaces in general topology
Zbl 0546.00022