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 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 ${\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}}$ 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 ${\textstyle r={\sqrt {(1+c)/(1-c)}}}$. 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 ${\textstyle r\geq {\sqrt {(1+c)/(1-c)}}}$. 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

Let ${\displaystyle X}$ be a topological space. Put ${\displaystyle X^{*}=X\cup \{\infty \},}$ and topologize ${\displaystyle X^{*}}$ by taking as open sets all the open sets in X together with all sets of the form ${\displaystyle V=(X\setminus C)\cup \{\infty \}}$ where C is closed and compact in X. Here, ${\displaystyle X\setminus C}$ denotes the complement of ${\displaystyle C}$ in ${\displaystyle X.}$ 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 space ${\displaystyle X^{*}}$ is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map ${\displaystyle c:X\to X^{*}.}$

The properties below 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.

See also

• Bohr compactification  – compact Hausdorff group associated to a topological groupPages displaying wikidata descriptions as a fallback
• Compact space  – Type of mathematical space
• Compactification (mathematics)  – Embedding a topological space into a compact space as a dense subset
• End (topology)  – in topology, the connected components of the “ideal boundary” of a spacePages displaying wikidata descriptions as a fallback
• Extended real number line  – Real numbers with +∞ and −∞ added
• Normal space  – topological space in which every pair of disjoint closed sets has disjoint open neighborhoodsPages displaying wikidata descriptions as a fallback
• Pointed set  – Basic concept in set theory
• Riemann sphere  – Model of the extended complex plane plus a point at infinity
• Stereographic projection  – Particular mapping that projects a sphere onto a plane
• Stone–Čech compactification  – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βXPages displaying wikidata descriptions as a fallback
• Wallman compactification  – A compactification of T₁ topological spaces

Notes

1. "General topology – Non-Hausdorff one-point compactifications".
2. Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN   0-387-96678-1 (See Chapter 11 for proof.)

References

• 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, S2CID   121699713
• Brown, Ronald (1973), "Sequentially proper maps and a sequential compactification", Journal of the London Mathematical Society , Series 2, 7 (3): 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", Encyclopedia of Mathematics , EMS Press
• 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