TY - JOUR
T1 - Separable quotients of free topological groups
AU - Leiderman, Arkady
AU - Tkachenko, Mikhail
N1 - 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 -