# Alexandroff extension

(Redirected from One-point 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 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, a much larger class of 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 ${\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}}$  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 ${\displaystyle (0,0,1)}$  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.

## Motivation

Let ${\displaystyle c:X\hookrightarrow Y}$  be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder ${\displaystyle \{\infty \}=Y\setminus c(X)}$ . 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 ${\displaystyle \infty }$  must be all sets obtained by adjoining ${\displaystyle \infty }$  to the image under c of a subset of X with compact complement.

## The Alexandroff extension

Put ${\displaystyle X^{*}=X\cup \{\infty \}}$ , and topologize ${\displaystyle X^{*}}$  by taking as open sets all the open subsets U of X together with all sets ${\displaystyle V=(X\setminus C)\cup \{\infty \}}$  where C is closed and compact in X. Here, ${\displaystyle X\setminus C}$  denotes setminus. Note that ${\displaystyle V}$  is an open neighborhood of ${\displaystyle \{\infty \}}$ , and thus, any open cover of ${\displaystyle \{\infty \}}$  will contain all except a compact subset ${\displaystyle C}$  of ${\displaystyle X^{*}}$ , implying that ${\displaystyle X^{*}}$  is compact (Kelley 1975, p. 150).

The inclusion map ${\displaystyle c:X\rightarrow X^{*}}$  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 ${\displaystyle X^{*}}$ .
• The space ${\displaystyle X^{*}}$  is compact.
• The image c(X) is dense in ${\displaystyle X^{*}}$ , if X is noncompact.
• The space ${\displaystyle X^{*}}$  is Hausdorff if and only if X is Hausdorff and locally compact.
• The space ${\displaystyle X^{*}}$  is T1 if and only if X is T1.

## The one-point compactification

In particular, the Alexandroff extension ${\displaystyle c:X\rightarrow X^{*}}$  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 ${\displaystyle X}$  is a compact Hausdorff space and ${\displaystyle p}$  is a limit point of ${\displaystyle X}$  (i.e. not an isolated point of ${\displaystyle X}$ ), ${\displaystyle X}$  is the Alexandroff compactification of ${\displaystyle X\setminus \{p\}}$ .

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set ${\displaystyle {\mathcal {C}}(X)}$  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 ${\displaystyle (X,\tau )}$  be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of ${\displaystyle X}$  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 ${\displaystyle X^{*}=X\cup \{\infty \}}$  a compact topology such that ${\displaystyle X}$  is dense in it and the subspace topology on ${\displaystyle X}$  induced from ${\displaystyle X^{*}}$  is the same as the original topology. The last compatibility condition on the topology automatically implies that ${\displaystyle X}$  is dense in ${\displaystyle X^{*}}$ , because ${\displaystyle X}$  is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map ${\displaystyle c:X\to X^{*}}$  is necessarily an open embedding, that is, ${\displaystyle X}$  must be open in ${\displaystyle X^{*}}$  and the topology on ${\displaystyle X^{*}}$  must contain every member of ${\displaystyle \tau }$ .[1] So the topology on ${\displaystyle X^{*}}$  is determined by the neighbourhoods of ${\displaystyle \infty }$ . Any neighborhood of ${\displaystyle \infty }$  is necessarily the complement in ${\displaystyle X^{*}}$  of a closed compact subset of ${\displaystyle X}$ , as previously discussed.

The topologies on ${\displaystyle X^{*}}$  that make it a compactification of ${\displaystyle X}$  are as follows:

• The Alexandroff extension of ${\displaystyle X}$  defined above. Here we take the complements of all closed compact subsets of ${\displaystyle X}$  as neighborhoods of ${\displaystyle \infty }$ . This is the largest topology that makes ${\displaystyle X^{*}}$  a one-point compactification of ${\displaystyle X}$ .
• The open extension topology. Here we add a single neighborhood of ${\displaystyle \infty }$ , namely the whole space ${\displaystyle X^{*}}$ . This is the smallest topology that makes ${\displaystyle X^{*}}$  a one-point compactification of ${\displaystyle X}$ .
• Any topology intermediate between the two topologies above. For neighborhoods of ${\displaystyle \infty }$  one has to pick a suitable subfamily of the complements of all closed compact subsets of ${\displaystyle X}$ ; 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 ${\displaystyle \{a_{n}\}}$  in a topological space ${\displaystyle X}$  converges to a point ${\displaystyle a}$  in ${\displaystyle X}$ , if and only if the map ${\displaystyle f\colon \mathbb {N} ^{*}\to X}$  given by ${\displaystyle f(n)=a_{n}}$  for ${\displaystyle n}$  in ${\displaystyle \mathbb {N} }$  and ${\displaystyle f(\infty )=a}$  is continuous. Here ${\displaystyle \mathbb {N} }$  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 ${\displaystyle \kappa }$  copies of the half-closed interval [0,1), that is, of ${\displaystyle [0,1)^{\kappa }}$ , is (homeomorphic to) ${\displaystyle [0,1]^{\kappa }}$ .
• 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 ${\displaystyle n}$  of copies of the interval (0,1) is a wedge of ${\displaystyle n}$  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 ${\displaystyle X}$  compact Hausdorff and ${\displaystyle C}$  any closed subset of ${\displaystyle X}$ , the one-point compactification of ${\displaystyle X\setminus C}$  is ${\displaystyle X/C}$ , where the forward slash denotes the quotient space.[2]
• If ${\displaystyle X}$  and ${\displaystyle Y}$  are locally compact Hausdorff, then ${\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}}$  where ${\displaystyle \wedge }$  is the smash product. Recall that the definition of the smash product:${\displaystyle A\wedge B=(A\times B)/(A\vee B)}$  where ${\displaystyle A\vee B}$  is the wedge sum, and again, / denotes the quotient space.[2]

### 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 ${\displaystyle c\colon X\rightarrow Y}$  and for which the morphisms from ${\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}}$  to ${\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}}$  are pairs of continuous maps ${\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}}$  such that ${\displaystyle f_{Y}\circ c_{1}=c_{2}\circ f_{X}}$ . In particular, homeomorphic spaces have isomorphic Alexandroff extensions.