Categorically closed topological groups (Q2275132)

Summary: Let \(\vec{\mathcal C}\) 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 \(\vec{\mathcal C}\) is called \(\vec{\mathcal C}\)-\textit{closed} if for each morphism \(\Phi\subset X \times Y\) in the category \(\vec{\mathcal C}\) 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 \(\vec{\mathcal C}\)-closed for various categories \(\vec{\mathcal C}\) of topologized semigroups.