Connectifying a topological space by adding one point

P. Alexandroff proved that a locally compact -space has a one-point compactification (obtained by adding a ``point at infinity'') if and only if it is non-compact. Also he asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in an attempt to answer Alexandroff's question (and in analogy with Alexandroff's theorem) we prove that in the class of -spaces ( ) 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 by assuming the set-theoretic assumption , and to the case 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 instead consider the collection of all one-point connectifications of a locally connected locally compact space in the class of -spaces ( ). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order structure determines the topology of all Stone-- ech remainders of components of the space.