World Library  
Flag as Inappropriate
Email this Article

Alexandroff compactification

Article Id: WHEBN0008835006
Reproduction Date:

Title: Alexandroff compactification  
Author: World Heritage Encyclopedia
Language: English
Subject: List of Russian people, Pavel Alexandrov, Modular curve
Publisher: World Heritage Encyclopedia

Alexandroff compactification

In mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematician Pavel Alexandrov.

More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any Tychonoff space, a much larger class of spaces.

Example: inverse stereographic projection

A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection S^{-1}: \mathbb{R}^2 \hookrightarrow S^2 is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point \infty = (0,0,1). Under the stereographic projection latitudinal circles z = c get mapped to planar circles r = \sqrt{\frac{1+c}{1-c}}. It follows that the deleted neighborhood basis of (1,0,0) given by the punctured spherical caps c \leq z < 1 corresponds to the complements of closed planar disks r \geq \sqrt{\frac{1+c}{1-c}}. More qualitatively, a neighborhood basis at \infty is furnished by the sets S^{-1}(\mathbb{R}^2 \setminus K) \cup \{ \infty \} as K ranges through the compact subsets of \mathbb{R}^2. This example already contains the key concepts of the general case.


Let c: X \hookrightarrow Y be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder \{ \infty \} = Y \setminus c(X). Then c(X) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff. Moreover, if X were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one point compactification the image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of \infty must be all sets obtained by adjoining \infty to the image under c of a subset of X with compact complement.

The Alexandroff extension

Let X be any topological space, and let \infty be any object which is not already an element of X. (In terms of formal set theory one could take, for example, \infty to be X itself, but it is not really necessary or helpful to be so specific.) Put X^* = X \cup \{\infty \}, and topologize X^* by taking as open sets all the open subsets U of X together with all subsets V which contain \infty and such that X \setminus V is closed and compact, (Kelley 1975, p. 150).

The inclusion map c: X \rightarrow X^* 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 X^*.
  • The space X^* is compact.
  • The image c(X) is dense in X^*, if X is noncompact.
  • The space X^* is Hausdorff if and only if X is Hausdorff and locally compact.

The one-point compactification

In particular, the Alexandroff extension c: X \rightarrow X^* is a compactification of X if and only if X is Hausdorff, noncompact and locally compact. In this case it is called the one-point compactification or Alexandroff compactification of X. Recall from the above discussion that any compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set \mathcal{C}(X) of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.

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 \kappa copies of the interval (0,1) is a wedge of \kappa circles.
  • The Alexandroff extension can be viewed as a functor from the category of topological spaces to the category whose objects are continuous maps c: X \rightarrow Y and for which the morphisms from c_1: X_1 \rightarrow Y_1 to c_2: X_2 \rightarrow Y_2 are pairs of continuous maps f_X: X_1 \rightarrow X_2, \ f_Y:

Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

See also


  • Template:Springer
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.