Let \({\mathcal P}\) be a cover of a space \(X\). Then \({\mathcal P}\) is a \(k\)-network for \(X\), if whenever \(K\subset U\) with \(K\) compact and \(U\) open in \(X,K\subset \bigcup{\mathcal P}'\subset U\) for some finite \({\mathcal P}'\subset{\mathcal P}\). A cover \({\mathcal C}\) of a space \(X\) is compact-countable if each compact subset of \(X\) meets only countably many elements of \({\mathcal C}\). Let \({\mathcal K}\) be a class of pseudo-open \(s\)-images of metric spaces; or \(k\)-spaces having a compact-countable closed \(k\)-network. Let \(\mathcal K'\) be a class of Fréchet spaces having a point-countable \(k\)-network; or point-\(G_\delta\) \(k\)-spaces having a compact-countable \(k\)-network. For spaces \(X\) and \(Y\in{\mathcal K}\) (or \({\mathcal K}'\)), this paper gives the following characterizations (A) and (B) for \(X\times Y\) to be a \(k\)-space. Here, a pair \((X,Y)\) of spaces \(X\) and \(Y\) fulfils the Tanaka condition if one of properties (a)\(\sim\)(c) holds: (a) \(X\) and \(Y\) are first countable spaces; (b) \(X\) or \(Y\) is a locally compact space; (c) \(X\) and \(Y\) are locally \(k_\omega\)-spaces. (A) For \(X,Y\in{\mathcal K}\), \(X\times Y\) is a \(k\)-space if and only if \((X,Y)\) fulfils the Tanaka condition. (B) The set-theoretic axiom \(BF(\omega_2)\) is false \(\iff\) For each \(X,Y\in{\mathcal K}'\), \(X\times Y\) is a \(k\)-space if and only if \((X,Y)\) fulfils the Tanaka condition. Also, it is shown that, for countably many spaces \(X_n\) in the class \({\mathcal K}\) (or \({\mathcal K}'\)), the same results for the countable product \(\prod X_n\) to be a \(k\)-space hold. \{Related matters to this paper are investigated synthetically by the reviewer [Products of \(k\)-spaces having point-countable \(k\)-networks, Topology Proc. 22, 305-329 (1997)]\}.