On countable coverings of word values in profinite groups

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w  -values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G)w(G) is locally finite. If G is a profinite group in which all w  -values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G)w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w  -values have finite order, then w(G)w(G) is locally finite and has finite exponent.