Examples concerning extensions of continuous functions (Q1880716)

The problems connected with continuous extension of continuous functions are considered. A space \(X\) is called to be a total extender for given space \(Y\) provided that every continuous function defined on a subset of \(X\) with values in \(Y\) admits a continuous extension over \(X\). It is known 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 \(Y\). In the paper under review it is shown that in the above statement one can not drop the hypothesis of metrizability of \(Y\) (leaving the assmption of compactness unchanged). However it is proved 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. A non-trivial example of such an \(X\) is given.