On non-topologizable semigroups

2024/05/27 by Gutik, Oleg
#22A15 #54C08 #54D10 #54D30 #54E52 #54H10 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2405.16992

Abstract

We find anti-isomorphic submonoids \mathscrC₊(a,b) and \mathscrC_-(a,b) of the bicyclic monoid \mathscrC(a,b) with the following properties: every Hausdorff left-continuous (right-continuous) topology on \mathscrC₊(a,b) (\mathscrC_-(a,b)) is discrete and there exists a compact Hausdorff topological monoid S which contains \mathscrC₊(a,b) (\mathscrC_-(a,b)) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology τ_p⁺ (τ_p^-) on the semigroup \mathscrC₊(a,b) (\mathscrC_-(a,b)) which is not left-continuous (right-continuous).

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