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

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.