Sequential rectifiable spaces of countable $cs^*$-character

TL;DR

The paper characterizes non-metrizable sequential rectifiable spaces with countable $cs^*$-character, showing they are either metrizable or decomposable into submetrizable $k_$-spaces, and proves they are submetrizable and paracompact.

Contribution

It provides a complete classification of such spaces, answering a question posed by Lin and Shen in 2011.

Findings

Non-metrizable spaces contain a clopen rectifiable submetrizable $k_$-subspace.

Spaces are either metrizable or decomposable into submetrizable $k_$-spaces.

Spaces are submetrizable and paracompact.

Abstract

We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_\omega$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen subspaces of $H$. This implies that each sequential rectifiable space with countable $cs^*$-character either is metrizable or else is a topological sum of submetrizable $k_\omega$-spaces. Consequently, $X$ is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.