Two problems concerning
F-spaces and F′-spaces are investigated. The first problem is to characterize those
F-spaces whose product with every P-space is an F-space. A new necessary condition
is obtained which is in fact a characterization of those F-spaces whose product with
any P-space with only one non-isolated point is an F-space. As a corollary an
example of a locally compact F-space and a P-space whose product is not an
F-space is obtained. The second problem is to verify a conjecture of Comfort,
Hindman and Negrepontis. It is shown that each weakly Lindelöf F′-space is an
F-space. Also, each zero-dimensional weakly Lindelöf F′-space is strongly
zero-dimensional.
