Alexandroff Extension - Further Examples

Further Examples

  • The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer.} with the order topology.
  • The one-point compactification of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse stereographic projection.
  • Since the closure of a connected subset is connected, the Alexandroff extension of a noncompact connected space is connected. However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of copies of the interval (0,1) is a wedge of circles.
  • The Alexandroff extension can be viewed as a functor from the category of topological spaces to the category whose objects are continuous maps and for which the morphisms from to are pairs of continuous maps f_X: X_1 \rightarrow X_2, \ f_Y: Y_1 \rightarrow Y_2 such that . In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

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