Necessity of the k-space hypothesis in Propositions 7 and 8

Ascertain whether the k-space hypothesis can be removed from Propositions 7 and 8; specifically, determine if (i) perfect preimages of C(K)-spaces (respectively Cs(K)-spaces) remain C(K)-spaces (respectively Cs(K)-spaces), and (ii) the topological sum E ⊕ E is a C(K)-space (respectively Cs(K)-space) for every C(K)-space E, when K is not assumed to consist of k-spaces.

Background

Proposition 7 establishes that if K is a class of k-spaces, then perfect preimages of C(K)-spaces (and Cs(K)-spaces) preserve the respective closed graph properties. Proposition 8 similarly shows that when K is a class of k-spaces and E is a C(K)-space, the topological sum E ⊕ E remains a C(K)-space, with an analogous statement for Cs(K)-spaces.

The authors explicitly note uncertainty about whether the requirement that K be composed of k-spaces is essential in these results. Removing this hypothesis would significantly strengthen the invariance properties of C(K)- and Cs(K)-spaces under perfect preimages and sums.