K-Spaces Property of Product Spaces

Let K be a class of spaces which are eigher a pseudo-open s-image of a metric space or ak-space having a compact-countable closed k-network. Let K′ be a class of spaces which are either aFréchet space with a point-countable k-network or a point-Gk-space having a compact-countablek-network. In this paper, we obtain some sufficient and necessary conditions that the products offinitely or countably many spaces in the class K or K′ are a k-space. The main results are thatTheorem A If X, Y ∈ K. Then X x Y is a k-space if and only if (X, Y) has the Tanaka’scondition.Theorem B The following are equivalent:(a) BF(w2) is false.(b) For each X, Y ∈ K′, X x Y is a k-space if and only if (X, Y) has the Tanaka’s condition.