Examples concerning extensions of continuous functions

Given a space Y, let us say that a space X is a total extender for Y provided-that every continuous map f : A --> Y defined on a subspace A of X admits a continuous extension (f) over tilde : X --> Y over X. The first author and Alberto Marcone proved that a space X is hereditarily extremally disconnected and hereditarily normal if and only if it is a total extender for every compact metrizable space Y, and asked whether the same result holds without any assumption of metrizability on Y. We demonstrate that a hereditarily extremally disconnected, hereditarily normal, non-collectionwise Hausdorff space X constructed by Kenneth Kunen is not a total extender for K, the one-point compactification of the discrete space of size omega(1). Under the assumption 2(omega0) = 2(omega1), we provide an example of a separable, hereditarily extremally disconnected, hereditarily normal space X that is not a total extender for K. Furthermore, using forcing we prove that, in the generic extension of a model of ZFC + MA(omega(1)), every first-countable separable space X of size omega(1) has a finer topology tau on X such that (X, tau) is still separable and fails to be a total extender for K. We also show that a hereditarily extremally disconnected, hereditarily separable space X satisfying some stronger form of hereditary normality (so-called structural normality) is a total extender for every compact Hausdorff space, and we give a non-trivial example of such an X. (C) 2004 Elsevier B.V. All rights reserved.