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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“The motive of science was the extension of man, on all sides, into Nature, till his hands should touch the stars, his eyes see through the earth, his ears understand the language of beast and bird, and the sense of the wind; and, through his sympathy, heaven and earth should talk with him. But that is not our science.”
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