Locally sigma-compact rectifiable spaces

摘要

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism phi : G x G -> G x G and an element e is an element of G such that pi(1) circle phi = pi(1). and for every x is an element of G, phi(x, x) = (x, e), where pi(1) : G x G -> G is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [2]). Then we prove that every locally sigma-compact rectifiable space with a bc-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each k(omega)-rectifiable space is rectifiable complete. (C) 2015 Elsevier B.V. All rights reserved.