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Submetrizability in paratopological groups. (English) Zbl 1300.54040

Given a paratopological group \(G\), i.e., a group with a \(T_0\)-topology such that the product map \(G\times G\to G\) is jointly continuous, it is natural to ask under which circumstances there exists a weaker topology on \(G\) such that \(G\) with this topology is metrizable (or even a metrizable topological group). In [Topology Appl. 159, No. 10–11, 2764–2773 (2012; Zbl 1276.54027)], F. Lin and C. Liu have shown that it does not suffice that every point in \(G\) is a \(G_\delta\)-set, thereby answering a question of A. Arhangel’skii and M. Tkachenko. Hence it is an interesting result of Xie and Lin that, if \(G\) is a feebly compact Hausdorff paratopological group with countable \(\pi\)-character, then there exists a weaker topology on \(G\) that turns \(G\) into a metrizable topological group. The same is shown to be true if \(G\) is a feebly compact paratopological group in which the identity \(e\) is a regular \(G_\delta\)-set.
Following M. Tkachenko [Topology Appl. 156, No. 7, 1298–1305 (2009; Zbl 1166.54016)], the Hausdorff number \(H_s(G)\) of a Hausdorff paratopological group \(G\) is the minimum cardinal number \(\kappa\) such that, for every neighborhood \(U\) of the identity \(e\), there exists a family \({\mathcal V}\) of neighborhoods of \(e\) such that \(\bigcap_{V\in{\mathcal V}} VV^{-1}\subset U\) and \(|{\mathcal V}|\leq\kappa\). Moreover, a paratopological group \(G\) is said to be 2-oscillating if, for every open neighborhood \(U\) of \(e\), there is an open neighborhood \(V\) of \(e\) such that \(V^{-1}V\subset UU^{-1}\). It is shown that if \(G\) is a 2-oscillating Hausdorff paratopological group with \(H_s(G)\cdot\psi(G)\leq \omega\), then there exists a weaker topology on \(G\) such that \(G\) with this topology is metrizable. If, in addition, \(G\) is \(\omega\)-balanced, then there exists a weaker topology on \(G\) that turns \(G\) into a metrizable topological group, where \(G\) is \(\omega\)-balanced if, for every open neighborhood \(U\) of \(e\), there exists a family \({\mathcal V}\) of open neighborhoods of \(e\) with \(|{\mathcal V}|\leq\omega\) such that, for every \(x\in G\), one can find a \(V\in{\mathcal V}\) satisfying \(xVx^{-1}\subset U\).
A paratopological group \(G\) is said to be \(\omega\)-narrow if, for every open neighborhood \(U\) of \(e\), there exists an \(A\in[G]^{\leq\omega}\) such that \(AU= G= UA\). The authors prove that if \(G\) is an \(\omega\)-narrow and \(\omega\)-balanced Hausdorff paratopological group with \(H_s(G)\cdots\psi(G)\leq\omega\), then there always exists a continuous bijection from \(G\) onto a second countable Hausdorff space.

MSC:

54E35 Metric spaces, metrizability
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54H11 Topological groups (topological aspects)
54H15 Transformation groups and semigroups (topological aspects)
20N99 Other generalizations of groups
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