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.

Characterizing categorically closed commutative semigroups

“…C-closed topological groups for various classes C were investigated by many authors including Arhangel'skii, Banakh, Choban, Dikranjan, Goto, Lukaśc and Uspenskij [1,2,5,8,15]. C-Closed topological semilattices were investigated by Gutik, Repovš, Stepp and the authors in [3,11,12,16,17].…”

“…The following characterization of C-closed commutative groups was obtained by the first author in [2].…”

“…. , f n } where f 0 = e. The Pigeonhole Principle implies that there exists k ≤ n and an infinite subset B ⊆ A such that {x 2 : x ∈ B} ⊆ H e if k = 0 and {x 2 : 2 we denote the set of all two-element subsets of B. Consider the function χ : [B] 2 → {0, .…”

Characterizing chain-compact and chain-finite topological semilattices

“…In 1969 Stepp [13,14] introduced e:TS-closed and h:TS-closed topological semigroups calling them maximal and absolutely maximal semigroups, respectively. The study h:TG-closed and p:TG-closed topological groups (called h-complete and hereditarily h-complete topological groups, respectiely) was initiated by Dikranjan and Tonolo [7] and continued by Dikranjan, Uspenskij [8], Lukaśc [11] and Banakh [1]. In [3,5,4,9] Hausdorff e:TS-closed (resp.…”

H-closed quasitopological groups

“…For instance, it holds for a group (G, τ ) for which there exists a σ-compact subgroup L of G such that G/L is periodic and there exists a group topology τ ′ ⊂ τ such that the Raǐkov completionĜ of the group (G, τ ′ ) is Baire (see [21,Proposition 17]). Also the conjecture is true if (G, τ ) is countable, or divisible, or characters of the group (G, τ ) separate its points and (G, τ ) isČech complete or periodic [6].…”