Categorically Closed Topological Groups

Abstract

Let 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 C → is called C → -closed if for each morphism Φ ⊂ X × Y in the category C → the image Φ ( X ) = { y ∈ Y : ∃ x ∈ X ( x , y ) ∈ Φ } is closed in Y. In the paper we survey existing and new results on topological groups, which are C → -closed for various categories C → of topologized semigroups.

Keywords

• topological group
• paratopological group
• topological semigroup
• absolutely closed topological group
• topological group of compact exponent