COMPACT SPACES REPRESENTABLE AS UNIONS OF NICE SUBSPACES

We present several results on compact Hausdorff spaces which can be represented as unions of nice subspaces. Some typical results are: If X is a compact Hausdorff space, and X = U(alpha < kappa)X(alpha), where each X(alpha) is kappa-refinable and PSI(X(alpha)) less-than-or-equal-to kappa, then (i) every nonempty G(kappa)-subset of X contains a point of character less-than-or-equal-to kappa, (ii) if x is-an-element-of X, chi(x, X) = mu > kappa and mu is regular, then there exists a discrete sequence {x(alpha): alpha < mu} in X such that x(alpha) --> x, (iii) if A is a nonclosed subset of X, then there exists a point x is-an-element-of X\A and a filter base F of subsets of A such that \F\ less-than-or-equal-to kappa and F --> x. We also show that if a compact Hausdorff space X is a union of countably many metrizable spaces, X has no isolated points and c(X) = omega0, then X is a compactification of the space of irrationals.