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 : 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^{omega_0} = 2^{omega_1}, 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.

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