On local compactness, pseudocompactness, and homogeneity

We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that w ( X ) ≤ n w ( X ) p c t ( X ) for any Hausdorff space X , where p c t ( X ) is the point-wise compactness type of X . This extends Arhangel′skiĭ's result that w ( X ) = n w ( X ) when X is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if X is a ccc locally compact noncompact space such that the one-point compactification of X is homogeneous and has character c , then X is pseudocompact. It follows that if X is either [ 0 , 1 ] c or 2 c and p ∈ X then X ﹨ { p } is pseudocompact.