A note on rectifiable spaces

Abstract

In this paper, we firstly discuss the question: Is l ∞ 2 homeomorphic to a rectifiable space or a paratopological group? And then, we mainly discuss locally compact rectifiable spaces, and show that a locally compact rectifiable space with the Souslin property is σ-compact, which gives an affirmative answer to A.V. Arhangel'skiǐ and M.M. Choban's question [On remainders of rectifiable spaces, Topology Appl., 157(2010), 789-799]. Next, we show that a rectifiable space X is strongly Fréchet-Urysohn if and only if X is an α 4sequential space. Moreover, we discuss the metrizabilities of rectifiable spaces, which gives a partial answer for a question posed in [18]. Finally, we consider the remainders of rectifiable spaces, which improve some results in [2,3, 6,19].

Sequential Rectifiable Spaces of Countable $$\mathrm {cs}^*$$ cs ∗ -Character

“…The following theorem generalizes Theorem 4 of [2], Lemma 4 of [3] and Theorem 3.1 of [14] to e-normal topological lops. Proof.…”

“…This result was extended in [3,Lemma 4] to arbitrary topological groups and to all normal rectifiable spaces in [14,Theorem 3.1]. In this section we will further generalize this Banakh's result to so-called e-normal topological lops, in particular, all topological lops of countable cs * -character.…”

“…In Section 4 we study properties of rectifiable spaces that contain closed copies of the metric fan M ω and the Fréchet-Urysohn fan S ω , and prove that such rectifiable spaces cannot simultaneously be sequential and have countable cs * -character. For normal topological groups this result was proved by Banakh [2], it was generalized to arbitrary topological groups by Banakh and Zdomskyy [3] and to normal rectifiable spaces by F. Lin, C. Liu, and S. Lin [14]. Section 5 contains Key Lemma 5.1, which is technically the most difficult result of the paper.…”

Locally σ-compact rectifiable spaces

“…The Sorgenfrey line G is a paratopological group with no rectification on G. Further, it is easy to see that each rectifiable space is homogeneous. Recently, the study of rectifiable spaces has become an interesting topic in topological algebra, see [1,2,[7][8][9][10][11][12][13]. In particular, the following theorem plays an important role in the study of rectifiable spaces.…”

Topologically subordered rectifiable spaces and compactifications

“…[27] Let G be a non-locally compact, paracompact rectifiable space, and Y = bG \ G have locally quasi-G δ -diagonal. Then G and bG are separable and metrizable.…”