The weight and Lindel\"of property in spaces and topological groups

TL;DR

This paper investigates bounds on the weight of topological spaces and groups, establishing new inequalities involving dense subspaces, Lindel"of $ ext{Sigma}$-spaces, and subspaces of separable spaces, with both upper bounds and counterexamples.

Contribution

It provides novel upper bounds for the weight of spaces and topological groups based on properties of dense subspaces and Lindel"of $ ext{Sigma}$-spaces, and constructs examples with maximal weight.

Findings

Bound $w(X)\

Upper bounds for weights of topological groups involving dense Lindel"of $ ext{Sigma}$-subgroups

Existence of subspaces with maximal weight in separable spaces

Abstract

We show that if $Y$ is a dense subspace of a Tychonoff space $X$, then $w(X)\leq nw(Y)^{Nag(Y)}$, where $Nag(Y)$ is the Nagami number of $Y$. In particular, if $Y$ is a Lindel\"of $\Sigma$-space, then $w(X)\leq nw(Y)^\omega\leq nw(X)^\omega$. 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 Lindel\"of $\Sigma$-space, then $w(H)=w(G)\leq \psi(G)^\omega$. Further, if a Lindel\"of $\Sigma$-space $X$ generates a dense subgroup of a topological group $H$, then $w(H)\leq 2^{\psi(X)}$. 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 $2^{2^{\mathfrak c}}$, where ${\mathfrak c}=2^\omega$. We prove on the one hand that if a regular Lindel\"of $\Sigma$-space $Y$ is a subspace of a separable Hausdorff space, then $w(Y)\leq \mathfrak c$, and the same conclusion holds for a Lindel\"of $P$-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 $w(G)=2^{2^{\mathfrak c}}$, i.e. has the maximal possible weight.