Products of uncountably many $k$-spaces

Abstract:

It is shown that if a product of nonempty spaces is a $k$-space then for each infinite cardinal $\mathfrak {n}$ some product of all but $\mathfrak {n}$ of the factors has each $\mathfrak {n}$-fold subproduct $\mathfrak {n} - {\aleph _0}$-compact (each $\mathfrak {n}$-fold open cover has a finite subcover). An example is given, for each regular $\mathfrak {n}$, of a space $X$ which is not $\mathfrak {n} - {\aleph _0}$-compact (so ${X^{{\mathfrak {n}^ + }}}$ is not a $k$-space) for which ${X^\mathfrak {n}}$ is a $k$-space.