Basic homogeneity in the class of zero-dimensional compact spaces (Q2469559)

A topological space is \(B\)-homogeneous if it has an open base every element of which can be mapped onto every other one by a homeomorphism of the whole space. A zero-dimensional space is \(h\)-homogeneous if any two clopen subsets are homeomorphic. In 1998 M. V. Matveev proved that \(B\)-homogeneity and \(h\)-homogeneity are equivalent notions in the class of zero-dimensional first-countable compact spaces. In the present paper it is shown that under the continuum hypothesis there is a compact zero-dimensional space which admits a base of pairwise homeomorphic clopen subsets but it is not an \(h\)-homogeneous space, partially answering a question of M. V. Matveev in 1998.