Assuming the Continuum Hypothesis (CH), we prove that there exists a perfectly normal compact topological space Z and a countable set E⊂Z such that Z∖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.