Some properties of topological groups related to compactness

In the present article four problems from the A.V. Arhangel'skii and M.G. Tkachenko's book [8] are examined. Theorem 2.5 affirms that for any uncountable cardinal τ there exists a zero-dimensional hereditarily paracompact non-metrizable Abelian topological group G of the weight τ1=sup{2m:m<τ} which has a linearly ordered compactification bG of countable dyadicity index. In this connection, in Section 2 we present some properties of continuous images of Tychonoff product of compact spaces of the fixed weight τ. These spaces are called τ-dyadic. By virtue of Corollary 3.3, if G is a non-metrizable topological group of pointwise countable type, then the space Ge=G∖{e} is not homeomorphic to a topological group. Section 3 contains also other results of that kind. In Section 4 some sufficient conditions are presented, under which the compact Gδ-subset of the quotient space G/H is a Dugundji space.