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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.

##### MSC:
 54D50 $$k$$-spaces 54B10 Product spaces in general topology
##### Keywords:
product of two k-spaces