The existence of one-point connectifications

TL;DR

This paper characterizes when locally connected spaces have one-point connectifications across various separation axioms, extending classical results and exploring their structure and uniqueness properties.

Contribution

It generalizes Alexandroff's theorem to higher separation axioms and metrizable/paracompact spaces, providing new characterizations and analyzing the structure of all such connectifications.

Findings

A locally connected space has a one-point connectification iff it has no compact component for certain separation axioms.

The collection of all one-point connectifications forms a compact, conditionally complete lattice.

The order-structure of this lattice determines the topology of Stone-Cech remainders.

Abstract

P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a "point at infinity") if and only if it is non-compact. He also asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in analogy with Alexandroff's theorem, we prove that in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We extend this theorem to the case $i=6$ by assuming some set-theoretic assumption, and to the case $i=2$ by slightly modifying its statement. We further extended the theorem by proving that a locally connected metrizable (resp. paracompact) space has a metrizable (resp. paracompact) one-point connectification if and only if it has no compact component. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We consider the collection of all one-point connectifications of a locally connected locally compact space in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order-structure determines the topology of all Stone-Cech remainders of components of the space.