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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 onepoint 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.
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Example: inverse stereographic projection
A geometrically appealing example of onepoint 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 is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point . Under the stereographic projection latitudinal circles z = c get mapped to planar circles . It follows that the deleted neighborhood basis of (1,0,0) 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.
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