One-point connectifications

TL;DR

This paper characterizes when locally connected, completely regular spaces have a one-point connectification, showing it occurs precisely when the space contains no compact components.

Contribution

It provides a complete characterization of spaces with one-point connectifications within the class of completely regular spaces.

Findings

Locally connected, completely regular spaces have a one-point connectification iff they lack compact components.

The result generalizes the classical Alexandroff compactification to connectifications.

It resolves a long-standing question of Alexandroff regarding connectifications.

Abstract

A space Y is called an extension of a space X if Y contains X as a dense subspace. An extension Y of X is called a one-point extension if Y-X is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact non-compact space has a one-point compactification (known as the Alexandroff compactification) obtained by adding a point at infinity. A locally connected disconnected space, however, may fail to have a one-point connectification. It is indeed a long standing question of Alexandroff to characterize spaces which have a one-point connectification. Here we prove that in the class of completely regular spaces, a locally connected space has a one-point connectification if and only if it contains no compact component.