Separable Quotients of Free Topological Groups. Issue 3 (29th September 2020)

• Record Type: Journal Article
• Authors: Leiderman, Arkady; Tkachenko, Mikhail
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 3(2020)
• Page Start: 610
• Page End: 623
• Publication Date: 2020-09-29
• Subjects: 22A05, -- 54D65, -- 54C10; free topological group, -- quotient, -- separable; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439519000699
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial ( i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$ -compact spaces, the class of connected locally connected spaces, and some others. We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.