Editing Alexandroff extension (section)

== Non-Hausdorff one-point compactifications ==
Let <math>(X,\tau)</math> be an arbitrary noncompact topological space.  One may want to determine all the compactifications (not necessarily Hausdorff) of <math>X</math> obtained by adding a single point, which could also be called ''one-point compactifications'' in this context.  
So one wants to determine all possible ways to give <math>X^*=X\cup\{\infty\}</math> a compact topology such that <math>X</math> is dense in it and the subspace topology on <math>X</math> induced from <math>X^*</math> is the same as the original topology.  The last compatibility condition on the topology automatically implies that <math>X</math> is dense in <math>X^*</math>, because <math>X</math> is not compact, so it cannot be closed in a compact space.
Also, it is a fact that the inclusion map <math>c:X\to X^*</math> is necessarily an [[open map|open]] embedding, that is, <math>X</math> must be open in <math>X^*</math> and the topology on <math>X^*</math> must contain every member
of <math>\tau</math>.<ref>{{Cite web|url=https://math.stackexchange.com/questions/3817485/non-hausdorff-one-point-compactifications|title=General topology – Non-Hausdorff one-point compactifications}}</ref>
So the topology on <math>X^*</math> is determined by the neighbourhoods of <math>\infty</math>.  Any neighborhood of <math>\infty</math> is necessarily the complement in <math>X^*</math> of a closed compact subset of <math>X</math>, as previously discussed.

The topologies on <math>X^*</math> that make it a compactification of <math>X</math> are as follows:
* The Alexandroff extension of <math>X</math> defined above.  Here we take the complements of all closed compact subsets of <math>X</math> as neighborhoods of <math>\infty</math>.  This is the largest topology that makes <math>X^*</math> a one-point compactification of <math>X</math>.
* The [[open extension topology]].  Here we add a single neighborhood of <math>\infty</math>, namely the whole space <math>X^*</math>.  This is the smallest topology that makes <math>X^*</math> a one-point compactification of <math>X</math>.
* Any topology intermediate between the two topologies above.  For neighborhoods of <math>\infty</math> one has to pick a suitable subfamily of the complements of all closed compact subsets of <math>X</math>; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.