  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.
