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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Famous quotes containing the word extension:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)