Submitted to Topology Proceedings CLOSED LOCALLY PATH-CONNECTED SUBSPACES OF FINITE-DIMENSIONAL GROUPS ARE LOCALLY COMPACT

Abstract. We prove that each closed locally continuum-connected subspace of a finite dimensional topological group is locally compact. This allows us to construct many 1-dimensional metrizable separable spaces that are not home-omorphic to closed subsets of finite-dimensional topological groups, which answers in negative a question of D.Shakhmatov. Another corollary is a characterization of Lie groups as finite-dimensional locally continuum-connected topological groups. For locally path connected topological groups this character-ization was proved by Gleason and Palais in 1957. 1