Generalizing dyadicity and Esenin-Vol'pin's Theorem

Abstract

For a pair (G,X), where G is a topological group and X is a subspace of G, we consider continuous mappings f of G onto a topological space Y such that f(G)=Y=f(X). If, in addition, X is compact (σ-compact), then Y is called a gc-image of G (respectively, a gσc-image of G). For example, all dyadic compacta are gc-images of compact topological groups. Below Esenin-Vol'pin's Theorem on metrizability of first-countable dyadic compacta [10] is extended, in several ways, to gc-images and gσc-images, and to certain generalizations of them. Essential new features of these generalizations are: the compactness assumption and the topological group assumption are completely separated, compactness is weakened to σ-compactness, and first-countability is replaced by the Gδ-points requirement. In particular, in Section 2 the following Theorem 2.2 is proved. Suppose that G is a topological group, X is a Lindelöf Σ subspace of G, and f is a continuous mapping of G onto a space Y such that f(G)=f(X)=Y. Suppose also that Z is a closed subset of Y such that every point of Z is a Gδ-point in Y. Then Z has a countable network. Hence, if in addition Z is compact, or a p-space, then Z is separable and metrizable. See also Theorems 3.6, 4.6, where the last statement is a generalization of the main result to the case of paratopological groups.