Alexandroff compactification

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.

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 ${\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}}$ is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point ${\displaystyle \infty =(0,0,1)}$. Under the stereographic projection latitudinal circles ${\displaystyle z=c}$ get mapped to planar circles ${\displaystyle r={\sqrt {(1+c)/(1-c)}}}$. It follows that the deleted neighborhood basis of ${\displaystyle (1,0,0)}$ given by the punctured spherical caps ${\displaystyle c\leq z<1}$ corresponds to the complements of closed planar disks ${\displaystyle r\geq {\sqrt {(1+c)/(1-c)}}}$. More qualitatively, a neighborhood basis at ${\displaystyle \infty }$ is furnished by the sets ${\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}}$ as K ranges through the compact subsets of ${\displaystyle \mathbb {R} ^{2}}$. This example already contains the key concepts of the general case.

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