One-point connectifications of regular spaces

Abstract

It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the Alexandroff compactification) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property $P$ and give conditions on $P$ which guarantee the space to have a regular one-point connectification with $P$.