Compactification of cut-point spaces

TL;DR

This paper explores the embedding and structural properties of cut-point spaces, demonstrating conditions under which they embed into dendrites, are weakly orderable, or densely embed into reducible continua, and discusses persistent cut points.

Contribution

It introduces new embedding results for cut-point spaces into dendrites and continua, and characterizes weak orderability via the Stone-Čech compactification.

Findings

Spaces with fewer than continuum non-cut points embed into dendrites.

Weak orderability of Tychonoff cut-point spaces is characterized by their Stone-Čech compactification.

Separable metrizable cut-point spaces can densely embed into reducible continua without cut points.

Abstract

We show that if $X$ is a separable locally compact Hausdorff connected space with fewer than $\mathfrak c$ non-cut points, then $X$ embeds into a dendrite $D\subseteq \mathbb R ^2$, and the set of non-cut points of $X$ is a nowhere dense $G_\delta$-set. We then prove a Tychonoff cut-point space $X$ is weakly orderable if and only if $\beta X$ is an irreducible continuum. Finally, we show every separable metrizable cut-point space densely embeds into a reducible continuum with no cut points. By contrast, there is a Tychonoff cut-point space each of whose compactifications has the same cut point. The example raises some questions about persistent cut points in Tychonoff spaces.