On certain compact topological spaces. (English) Zbl 0870.54025

Summary: A compact topological space \(K\) is in the class \({\mathcal A}\) if it is homeomorphic to a subspace \(H\) of \([0,1]^I\), for some set of indexes \(I\), such that, if \(L\) is the subset of \(H\) consisting of all \(\{x_i:i\in I\}\) with \(x_i=0\) except for a countable number of \(i\)’s, then \(L\) is dense in \(H\). In this paper we show that the class \({\mathcal A}\) of compact spaces is not stable under continuous maps. This solves a problem posed by R. Deville, G. Godefroy and V. Zizler [Smoothness and renormings in Banach spaces (1993; Zbl 0782.46019)].


54D30 Compactness


Zbl 0782.46019
Full Text: EuDML