Alexandroff Extension - The Alexandroff Extension

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:

    Predatory capitalism created a complex industrial system and an advanced technology; it permitted a considerable extension of democratic practice and fostered certain liberal values, but within limits that are now being pressed and must be overcome. It is not a fit system for the mid- twentieth century.
    Noam Chomsky (b. 1928)