The weight and Lindelöf property in spaces and topological groups

Abstract

We show that if Y is a dense subspace of a Tychonoff space X, then w(X)≤nw(Y)Nag(Y), where Nag(Y) is the Nagami number of Y. In particular, if Y is a Lindelöf Σ-space, then w(X)≤nw(Y)ω≤nw(X)ω.Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense subgroup G such that G is a Lindelöf Σ-space, then w(H)=w(G)≤ψ(G)ω. Further, if a Lindelöf Σ-space X generates a dense subgroup of a topological group H, then w(H)≤2ψ(X).Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space X can be as big as 22c. We prove on the one hand that if a regular Lindelöf Σ-space Y is a subspace of a separable Hausdorff space, then w(Y)≤2ω, and the same conclusion holds for a Lindelöf P-space Y. On the other hand, we present an example of a countably compact topological Abelian group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w(G)=22c, i.e. G has the maximal possible weight.