Metrizability of GO-spaces and topological groups (Q2763921)

The author presents some metrizability theorems on generalized ordered spaces (that is, subspaces of linearly ordered topological spaces) and topological groups. The main tools are topologies determined by covers and the notion of \(wcs^{\star}\)-network. We recall that a topological space \((X,\mathcal{T})\) {is determined by a cover} \(\mathcal{C}\) if a subset \(F\subset X\) is closed in \((X,\mathcal{T})\) if and only if \(F\cap C\) is closed in \(C\) with the induced topology for every \(C\in \mathcal{C}\). For instance, a \(k\)-space is a topological space determined by a cover of compact subsets and a sequential space is a topological space determined by a cover of metric compact subsets. On the other hand, a cover \({\mathcal C}\) is \textit{a} \(wcs^{\star}\)-network for \((X,\mathcal{T})\) if whenever \(L\) is a sequence converging to a point \(x\in X\) and \(U\) is a neighborhood of \(x\), some \(P\in \mathcal{P}\) is contained in \(U\) and contains the sequence \(L\) frequently. As an example of the results obtained we have the following: (1) Let \((X,\mathcal{T})\) be a topological space determined by an increasing (not necessarily closed) countable cover \(\{X_{n}:n\in \mathbb{N}\}\). Then the following assertions hold: (i) If \((X,\mathcal{T})\) is a generalized ordered space, then \((X,\mathcal{T})\) is metrizable. Moreover, if each \(X_{n}\) is a strongly zero-dimensional space, then \((X,\mathcal{T})\) is a strongly zero-dimensional metrizable space if and only if it is a generalized ordered space if and only if it is a linearly ordered topological space, and (ii) if \((X,\mathcal{T})\) is a topological group, then it is either discrete or the topological sum of the real lines or the topological sum of the Cantor sets; (2) every generalized ordered space with a \(\sigma\)-locally countable \(wcs^{\star}\)-network is metrizable.