The Alexandroff Extension
Let X be any topological space, and let be any object which is not already an element of X. (In terms of formal set theory one could take, for example, to be X itself, but it is not really necessary or helpful to be so specific.) Put, and topologize by taking as open sets all the open subsets U of X together with all subsets V which contain and such that is closed and compact, (Kelley 1975, p. 150).
The inclusion map is called the Alexandroff extension of X (Willard, 19A).
The above properties all follow easily from the above discussion:
- The map c is continuous and open: it embeds X as an open subset of .
- The space is compact.
- The image c(X) is dense in, if X is noncompact.
- The space is Hausdorff if and only if X is Hausdorff and locally compact.
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