Weak C-embedding and P-embedding, and product spaces

Arhangelʹskiı̆ defines in [Topology Appl. 70 (1996) 87–99] a subspace Y of a topological space X to be weakly C-embedded in X if for every real-valued continuous function f on Y there exists a real-valued function on X which is an extension of f and continuous at each point of Y. In this paper, we prove that Y is weakly C-embedded in X if and only if for every pair G0, G1 of disjoint cozero-sets of Y there exist disjoint open subsets H0 and H1 of X such that Gi⊂Hi (i=0,1). As applications, our characterization gives answers with simple observations to problems posed by Arhangelʹskiı̆, which were recently solved by Bella and Yaschenko [Proc. Amer. Math. Soc. 127 (1999) 907–913], Costantini and Marcone [Topology Appl. 103 (2000) 131–153] and Matveev et al. [Topology Appl. 93 (1999) 121–129]. Moreover, suggested by weak C-embedding, we define a notion of weak P-embedding, more precisely, weak Pγ-embedding for an infinite cardinal γ, which is an extension of P-embedding. Using this notion we describe weak C-embedding for product spaces with a compact factor.