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.
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 (but provides an embedding exactly for Tychonoff spaces).
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.
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{(1+c)/(1-c)}.
It follows that the deleted neighborhood basis of (0,0,1) given by the punctured spherical caps c \leq z < 1 corresponds to the complements of closed planar disks r \geq \sqrt{(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.
Motivation
Let c: X \hookrightarrow Y be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder \{ \infty \} = Y \setminus c(X).
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 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 \infty must be all sets obtained by adjoining \infty to the image under c of a subset of X with compact complement.
The Alexandroff extension
Put X^* = X \cup \{\infty \}, and topologize X^* by taking as open sets all the open subsets U of X together with all sets V = (X \setminus C) \cup \{\infty \} where C is closed and compact in X.  Here, X \setminus C denotes setminus.
Note that V is an open neighborhood of \{\infty \}, and thus, any open cover of  \{\infty \} will contain all except a compact subset C of X^*, implying that X^* is compact .
The inclusion map c: X \rightarrow X^* 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 X^*.
The space X^* is compact.
The image c(X) is dense in X^*, if X is noncompact.
The space X^* is Hausdorff if and only if X is Hausdorff and locally compact.
The space X^* is T1 if and only if X is T1.
The one-point compactification
In particular, the Alexandroff extension c: X \rightarrow X^* 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 Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.
In particular, if X is a compact Hausdorff space and p is a limit point of X (i.e. not an isolated point of X), X is the Alexandroff compactification of X\setminus\{p\}.
Let X be any noncompact Tychonoff space.
Under the natural partial ordering on the set \mathcal{C}(X) 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.
Non-Hausdorff one-point compactifications
Let (X,\tau) be an arbitrary noncompact topological space.
One may want to determine all the compactifications (not necessarily Hausdorff) of X obtained by adding a single point, which could also be called one-point compactifications in this context.
So one wants to determine all possible ways to give X^*=X\cup\{\infty\} a compact topology such that X is dense in it and the subspace topology on X induced from X^* is the same as the original topology.
The last compatibility condition on the topology automatically implies that X is dense in X^*, because X is not compact, so it cannot be closed in a compact space.
Also, it is a fact that the inclusion map c:X\to X^* is necessarily an open embedding, that is, X must be open in X^* and the topology on X^* must contain every member of \tau.
So the topology on X^* is determined by the neighbourhoods of \infty.
Any neighborhood of \infty is necessarily the complement in X^* of a closed compact subset of X, as previously discussed.
The topologies on X^* that make it a compactification of X are as follows:
The Alexandroff extension of X defined above.
Here we take the complements of all closed compact subsets of X as neighborhoods of \infty.
This is the largest topology that makes X^* a one-point compactification of X.
The open extension topology.
Here we add a single neighborhood of \infty, namely the whole space X^*.
This is the smallest topology that makes X^* a one-point compactification of X.
Any topology intermediate between the two topologies above.
For neighborhoods of \infty one has to pick a suitable subfamily of the complements of all closed compact subsets of X; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.
Further examples
Compactifications of discrete spaces
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 \{a_n\} in a topological space X converges to a point a in X, if and only if the map f\colon\mathbb N^*\to X given by f(n) = a_n for n in \mathbb N and f(\infty) = a is continuous.
Here \mathbb N 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 spaces
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 \kappa copies of the half-closed interval [0,1), that is, of [0,1)^\kappa, is (homeomorphic to) [0,1]^\kappa.
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 a finite number n of copies of the interval (0,1) is a wedge of n circles.
The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is the Hawaiian earring.
This is different from the wedge of countably many circles, which is not compact.
Given X compact Hausdorff and C any closed subset of X, the one-point compactification of X\setminus C is X/C, where the forward slash denotes the quotient space.Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag  (See Chapter 11 for proof.)
If X and Y are locally compact Hausdorff, then (X\times Y)^* = X^* \wedge Y^* where \wedge is the smash product.
Recall that the definition of the smash product:A\wedge B = (A \times B) / (A \vee B) where A \vee B is the wedge sum, and again, / denotes the quotient space.
As a functor
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 c\colon X \rightarrow Y and for which the morphisms from c_1\colon X_1 \rightarrow Y_1 to c_2\colon X_2 \rightarrow Y_2 are pairs of continuous maps f_X\colon X_1 \rightarrow X_2, \ f_Y\colon   Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X.
In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
See also
Notes
References
