
In 1924, P. Alexandroff proved that a locally compact $T_2$space has a $T_2$ onepoint compactification (obtained by adding a ``point at infinity'') if and only if it is noncompact. He also asked for characterizations of spaces which have onepoint connectifications. Here, we consider onepoint connectifications, and in analogy with Alexandroff's theorem, we show that in the realm of $T_i$spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a onepoint connectification if and only if it has no compact component. We explain how this theorem may be extended to the case $i=2$ and to the case $i=6$ under the settheoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$. Contrary to the case of the onepoint compactification, a onepoint connectification, if exists, may not be unique. We consider the collection of all onepoint connectifications of a locally connected locally compact space in the realm of $T_i$spaces ($i=3\frac{1}{2},4,5$). This collection, naturally partially ordered, is a compact conditionally complete lattice whose orderstructure is rich enough to determine the topology of all StoneČech remainders of components of the space.

