Open AccessAn example of a non-Borel locally-connected finite-dimensional topological group
Author(s) -
Iryna Banakh,
Тарас Банах,
Myroslava Vovk
Publication year - 2017
Publication title -
karpatsʹkì matematičnì publìkacìï
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.63
H-Index - 4
eISSN - 2313-0210
pISSN - 2075-9827
DOI - 10.15330/cmp.9.1.3-5
Subject(s) - mathematics , social connectedness , path (computing) , dimension (graph theory) , group (periodic table) , simply connected space , combinatorics , topology (electrical circuits) , fundamental group , connected space , pure mathematics , discrete mathematics , topological space , psychology , chemistry , organic chemistry , computer science , psychotherapist , programming language
According to a classical theorem of Gleason and Montgomery, every finite-dimensional locally path-connected topological group is a Lie group. In the paper for every $n\ge 2$ we construct a locally connected subgroup $G\subset{\mathbb R}^{n+1}$ of dimension $\dim(G)=n$, which is not locally compact. This answers a question posed by S. Maillot on MathOverflow and shows that the local path-connectedness in the result of Gleason and Montgomery can not be weakened to the local connectedness.