Alexandroff extension

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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 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.

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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.

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