Remainders of products, topological groups and Cp-spaces

Abstract

Abstract Let X = ∏ i ∈ I X i be a product of non-compact spaces. We show that if | I | > ω , then the remainder Y = b X ∖ X is pseudocompact, for any compactification bX of X. In fact, this theorem follows from a more general result about spaces with an ω-directed lattice of d-open mappings. Under the additional assumption that the space X has countable cellularity, we prove that the remainder Y is C-embedded in bX and that β Y = b X . We apply these results to the remainders of topological groups and spaces of continuous functions with the pointwise convergence topology. For example, we prove that if X is an uncountable space and G is a non-compact topological group, then every remainder of C p ( X , G ) is pseudocompact provided C p ( X , G ) is dense in G X .