Necessary and sufficient conditions for products of \(k\)-spaces. (English) Zbl 0727.54012

The product of two k-spaces need not be a k-space, even if one of the factors is as nice as a separable metric space. The finding of necessary and sufficient conditions for the product of two k-spaces to be a k-space is a complicated process. This paper gives a number of such conditions, some of which involve, or are equivalent to, a set-theoretic assumption. Here is an example of a theorem from the paper giving one of the latter kind of condition. If X and Y are dominated by spaces which are closed images of metric spaces, then the statement that the cardinal number b is not greater than or equal to \(\omega_ 2\) is equivalent to the statement that \(X\times Y\) is a k-space if and only if one of the following holds: X or Y is locally compact; X and Y are metrizable; X and Y are locally \(k_{\omega}\). The author raises several questions, one of which is whether the equivalence in the above statement is true whenever X and Y are dominated by compact spaces. In the last section, conditions are given for \(X^{\omega}\) to be a k-space.


54D50 \(k\)-spaces
54B10 Product spaces in general topology