Alexandroff extension

Alexandroff extension

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.

Contents

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. Therefore 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.

Motivation

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 κ copies of the interval (0,1) is a wedge of κ 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

References

  • P.S. Alexandroff (1924), "Über die Metrisation der im Kleinen kompakten topologischen Räume", Math. Ann. 92 (3-4): 294–301, doi:10.1007/BF01448011 
  • Ronald Brown (1973), "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7: 515-522 

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