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In the 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 topological space, a much larger class of spaces.

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Example: inverse stereographic projectionEdit

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   is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point  . Under the stereographic projection latitudinal circles   get mapped to planar circles  . It follows that the deleted neighborhood basis of   given by the punctured spherical caps   corresponds to the complements of closed planar disks  . More qualitatively, a neighborhood basis at   is furnished by the sets   as K ranges through the compact subsets of  . This example already contains the key concepts of the general case.

MotivationEdit

Let   be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder  . 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   must be all sets obtained by adjoining   to the image under c of a subset of X with compact complement.

The Alexandroff extensionEdit

Put  , and topologize   by taking as open sets all the open subsets U of X together with all sets   where C is closed and compact in X. Here,   denotes setminus. Note that   is an open neighborhood of  , and thus, any open cover of   will contain all except a compact subset   of  , implying that   is compact (Kelley 1975, p. 150).

The inclusion map   is called the Alexandroff extension of X (Willard, 19A).

The properties below all follow 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.

The one-point compactificationEdit

In particular, the Alexandroff extension   is a Hausdorff 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   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 examplesEdit

Compactifications of discrete spacesEdit

  • 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.
  • A sequence   in a topological space   converges to a point   in  , if and only if the map   given by   for   in   and   is continuous. Here   has the discrete topology.
  • Polyadic spaces are defined as topological spaces that are the continuous image of the power of a one-point compactification of a discrete, locally compact Hausdorff space.

Compactifications of continuous spacesEdit

  • 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.
  • The one-point compactification of the product of   copies of the half-closed interval [0,1), that is, of  , is (homeomorphic to)  .
  • 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.
  • Given   compact Hausdorff and   any closed subset of  , the one-point compactification of   is  , where the forward slash denotes the quotient space.[1]
  • If   and   are locally compact Hausdorff, then   where   is the smash product. Recall that the definition of the smash product:  where   is the wedge sum, and again, / denotes the quotient space.[1]

Compactification of function spacesEdit

  • The space of continuous functions   on a locally compact Hausdorff space   is locally compact but can be made compact if and only if we include the single point   for all  .

As a functorEdit

The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps   and for which the morphisms from   to   are pairs of continuous maps   such that  . In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

See alsoEdit

ReferencesEdit

  1. ^ a b Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)