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 ) N ag (Y ) Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense sub-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 2 2 c . 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 ) ≤ 2 ω , and the same conclusion holds for a Lindelöf P -space Y . On the other hand, we present an example of a countably compact topological group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w(G) = 2 2 c , i.e. has the maximal possible weight.