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.
Read more about this topic: Alexandroff Extension
Famous quotes containing the word extension:
“The medium is the message. This is merely to say that the personal and social consequences of any medium—that is, of any extension of ourselves—result from the new scale that is introduced into our affairs by each extension of ourselves, or by any new technology.”
—Marshall McLuhan (1911–1980)