Some properties of rectifiable spaces (Q1673581)

In this paper, the authors investigate some properties of rectifiable spaces. The following main results are proved: (1)\, let \(G\) be a rectifiable space and \(F\) be a non-empty compact \(G_{\delta}\)-set in \(G\). Then there exists a \(G_{\delta}\)-set \(P\) in \(G\) such that \(e\in P\) and \(FP\subset F\); (2)\, let \(H\) be a dense rectifiable subspace of a rectifiable space \(G\). If \(H\) is a \(P\)-space, then \(G\) is a \(P\)-space; (3)\, a \(\kappa\)-Fréchet-Urysohn, biradial rectifiable space \(G\) is metrizable. These results are used to generalize some results of topological groups and rectifiable spaces. Moreover, the authors prove that every first-countable cosmic rectifiable space is second-countable.