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 after the Russian mathematician Pavel Alexandroff.
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 (but provides an embedding exactly for Tychonoff 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.
is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point
Under the stereographic projection latitudinal circles
It follows that the deleted neighborhood basis of
corresponds to the complements of closed planar disks
This example already contains the key concepts of the general case.
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.
Thus a space can only admit a Hausdorff 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
to the image under c of a subset of X with compact complement.
is called the Alexandroff extension of X (Willard, 19A).
The properties below follow from the above discussion: In particular, the Alexandroff extension
Recall from the above discussion that any Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.
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.
be an arbitrary noncompact topological space.
One may want to determine all the compactifications (not necessarily Hausdorff) of
obtained by adding a single point, which could also be called one-point compactifications in this context.
The last compatibility condition on the topology automatically implies that
are as follows: 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
In particular, homeomorphic spaces have isomorphic Alexandroff extensions.