Solution of Ponomarev's problem of condensation onto compact sets

Assuming the Continuum Hypothesis (CH), we prove that there exists a perfectly normal compact topological space $Z$ and a countable set $E\subset Z$ such that $Z\setminus E$ does not condense onto any compact set. The space $Z$ enables us to answer in the negative (under CH) the following problem of Ponomarev: Is each perfectly normal compact set an $a$-space? We also prove that the product of $a$-spaces need not be an $a$-space.