Connectifying a space by adding one point

TL;DR

This paper explores the conditions under which a locally connected normal space can be extended by adding a single point to become connected, paralleling classical compactification results.

Contribution

It establishes a necessary and sufficient condition for the existence of a one-point connectification in locally connected normal spaces.

Findings

A locally connected normal space has a one-point connectification iff it has no compact component.

The result parallels the classical one-point compactification theorem.

Provides a new criterion for connectification in topological spaces.

Abstract

It is a classical theorem of Alexandroff that a locally compact Hausdorff space has a one-point Hausdorff compactification if and only if it is non-compact. The one-point Hausdorff compactification is indeed obtained by adding the so called "point at infinity." Here we consider the analogous problem of existence of a one-point connectification, and keeping analogy, we prove that a locally connected normal space has a one-point normal connectification if and only if it has no compact component.