Categorically Closed Topological Groups

Author:

Banakh TarasORCID

Abstract

Let \({\overset{\rightarrow}{\mathcal{C}} }\) be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category \({\overset{\rightarrow}{\mathcal{C}} }\) is called \({\overset{\rightarrow}{\mathcal{C}} }\)-closed if for each morphism \({\Phi\subset X\times Y}\) in the category \({\overset{\rightarrow}{\mathcal{C}} }\) the image \({\Phi(X)=\{y\in Y:\exists x\in X\;(x,y)\in\Phi\}}\) is closed in Y. In the paper we survey existing and new results on topological groups, which are \({\overset{\rightarrow}{\mathcal{C}} }\)-closed for various categories \({\overset{\rightarrow}{\mathcal{C}} }\) of topologized semigroups.

Publisher

MDPI AG

Subject

Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis

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