TY - JOUR

T1 - Separable quotients of free topological groups

AU - Leiderman, Arkady

AU - Tkachenko, Mikhail

N1 - Funding Information:
Received by the editors May 31, 2019; revised November 18, 2019. Published online on Cambridge Core November 29, 2019. Author M. T. gratefully acknowledges the financial support received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit in May, 2019. AMS subject classification: 22A05, 54D65, 54C10. Keywords: free topological group, quotient, separable.
Publisher Copyright:
© 2020 Cambridge University Press. All rights reserved.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - We study the following problem: For which Tychonoò 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? he 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 σ-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.

AB - We study the following problem: For which Tychonoò 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? he 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 σ-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.

KW - Free topological group

KW - Quotient

KW - Separable

UR - http://www.scopus.com/inward/record.url?scp=85084223305&partnerID=8YFLogxK

U2 - 10.4153/S0008439519000699

DO - 10.4153/S0008439519000699

M3 - Article

AN - SCOPUS:85084223305

SN - 0008-4395

VL - 63

SP - 610

EP - 623

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

IS - 3

ER -