Alexandroff extension

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.

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 $(1,0,0)$ 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 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.(Kelley 1975, p. 150)

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 one-point compactification

In particular, the Alexandroff extension $c:X\rightarrow X^{*}$ is a 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 compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.

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.

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 $\kappa$ copies of the interval (0,1) is a wedge of $\kappa$ circles.
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.^[1]
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.^[1]

Compactification of function spaces

The space of continuous functions $C\left(\Omega \right)$ on a locally compact Hausdorff space $\Omega$ is locally compact but can be made compact if and only if we include the single point $f(x)=1$ for all $x$ .

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.

References

1 2 Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)

Alexandroff, Pavel S. (1924), "Über die Metrisation der im Kleinen kompakten topologischen Räume", Mathematische Annalen, 92 (3–4): 294–301, doi:10.1007/BF01448011, JFM 50.0128.04
Brown, Ronald (1973), "Sequentially proper maps and a sequential compactification", Journal of the London Mathematical Society, Series 2, 7: 515–522, doi:10.1112/jlms/s2-7.3.515, Zbl 0269.54015

Engelking, Ryszard (1989), General Topology, Helderman Verlag Berlin, ISBN 978-0-201-08707-9, MR 1039321
Fedorchuk, V.V. (2001) [1994], "Aleksandrov compactification", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Kelley, John L. (1975), General Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90125-1, MR 0370454
Munkres, James (1999), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2, Zbl 0951.54001
Willard, Stephen (1970), General Topology, Addison-Wesley, ISBN 3-88538-006-4, MR 0264581, Zbl 0205.26601

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[rotman-1] 1 2 Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)