On one-point metrizable extensions of locally compact metrizable spaces

TL;DR

This paper characterizes one-point metrizable extensions of locally compact metrizable spaces using zero-sets in the Stone-Čech compactification, and explores their order structure and cardinality.

Contribution

It provides a complete characterization of the image of the extension map and analyzes the structure and size of these extension sets.

Findings

Elements of the extension image are exactly non-empty zero-sets missing X.

Locally compact elements correspond to clopen zero-sets in βX\X.

Results include bounds on the cardinality of extension sets.

Abstract

For a non-compact metrizable space $X$, let ${\mathcal E}(X)$ be the set of all one-point metrizable extensions of $X$, and when $X$ is locally compact, let ${\mathcal E}_K(X)$ denote the set of all locally compact elements of ${\mathcal E}(X)$ and $\lambda: {\mathcal E}(X)\rightarrow{\mathcal Z}(\beta X\backslash X)$ be the order-anti-isomorphism (onto its image) defined in: [HJW] M. Henriksen, L. Janos and R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolinae 46 (2005), 105-123. By definition $\lambda(Y)= \bigcap_{n<\omega}cl_{\beta X}(U_n\cap X)\backslash X$, where $Y=X\cup\{p\}\in{\mathcal E}(X)$ and $\{U_n\}_{n<\omega}$ is an open base at $p$ in $Y$. Answering the question of [HJW], we characterize the elements of the image of $\lambda$ as exactly those non-empty zero-sets of $\beta X$ which miss $X$, and the elements of the image of ${\mathcal E}_K(X)$ under $\lambda $, as those which are moreover clopen in $\beta X\backslash X$. We then study the relation between ${\mathcal E}(X)$ and ${\mathcal E}_K(X)$ and their order structures, and introduce a subset ${\mathcal E}_S(X)$ of ${\mathcal E}(X)$. We conclude with some theorems on the cardinality of the sets ${\mathcal E}(X)$ and ${\mathcal E}_K(X)$, and some open questions.