In themathematical field oftopology, theAlexandroff extension is a way to extend a noncompacttopological space by adjoining a single point in such a way that the resulting space iscompact. It is named after the Russian mathematicianPavel Alexandroff.More precisely, letX be a topological space. Then the Alexandroff extension ofX is a certain compact spaceX* together with anopenembeddingc : X → X* such that the complement ofX inX* consists of a single point, typically denoted ∞. The mapc is a Hausdorffcompactification if and only ifX is alocally compact, noncompactHausdorff space. For such spaces the Alexandroff extension is called theone-point compactification orAlexandroff 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 theStone–Čech compactification which exists for anytopological space (butprovides an embedding exactly forTychonoff spaces).

A geometrically appealing example of one-point compactification is given by the inversestereographic projection. Recall that the stereographic projectionS 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${\displaystyle \mathrm{\infty }=(0,0,1)}$. Under the stereographic projection latitudinal circles${\displaystyle z=c}$ get mapped to planar circles$r=\sqrt{(1+c)/(1-c)}$. It follows that the deleted neighborhood basis of${\displaystyle (0,0,1)}$ given by the punctured spherical caps corresponds to the complements of closed planar disks$r\ge \sqrt{(1+c)/(1-c)}$. More qualitatively, a neighborhood basis at${\displaystyle \mathrm{\infty }}$ is furnished by the sets${\displaystyle S^{-1}({\mathbb{R}}^2\setminus K)\cup \{\mathrm{\infty }\}}$ asK ranges through the compact subsets of${\displaystyle {\mathbb{R}}^2}$. This example already contains the key concepts of the general case.

Let${\displaystyle c:X\hookrightarrow Y}$ be an embedding from a topological spaceX to a compact Hausdorff topological spaceY, with dense image and one-point remainder${\displaystyle \{\mathrm{\infty }\}=Y\setminus c(X)}$. Thenc(X) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimageX is also locally compact Hausdorff. Moreover, ifX were compact thenc(X) would be closed inY 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 forx inX gives a neighborhood basis forc(x) inc(X), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of${\displaystyle \mathrm{\infty }}$ must be all sets obtained by adjoining${\displaystyle \mathrm{\infty }}$ to the image underc of a subset ofX with compact complement.

Let${\displaystyle X}$ be a topological space. Put${\displaystyle X^{\ast }=X\cup \{\mathrm{\infty }\},}$ and topologize${\displaystyle X^{\ast }}$ by taking as open sets all the open sets inX together with all sets of the form${\displaystyle V=(X\setminus C)\cup \{\mathrm{\infty }\}}$ whereC is closed and compact inX. Here,${\displaystyle X\setminus C}$ denotes the complement of${\displaystyle C}$ in${\displaystyle X.}$ Note that${\displaystyle V}$ is an open neighborhood of${\displaystyle \mathrm{\infty },}$ and thus any open cover of${\displaystyle \{\mathrm{\infty }\}}$ will contain all except a compact subset${\displaystyle C}$ of${\displaystyle X^{\ast },}$ implying that${\displaystyle X^{\ast }}$ is compact (Kelley 1975, p. 150).

The space${\displaystyle X^{\ast }}$ is called theAlexandroff extension ofX (Willard, 19A). Sometimes the same name is used for the inclusion map${\displaystyle c:X\to X^{\ast }.}$

The properties below follow from the above discussion:

• The mapc is continuous and open: it embedsX as an open subset of${\displaystyle X^{\ast }}$.
• The space${\displaystyle X^{\ast }}$ is compact.
• The imagec(X) is dense in${\displaystyle X^{\ast }}$, ifX is noncompact.
• The space${\displaystyle X^{\ast }}$ isHausdorff if and only ifX is Hausdorff andlocally compact.
• The space${\displaystyle X^{\ast }}$ isT₁ if and only ifX is T₁.

In particular, the Alexandroff extension${\displaystyle c:X\to X^{\ast }}$ is a Hausdorff compactification ofX if and only ifX is Hausdorff, noncompact and locally compact. In this case it is called theone-point compactification orAlexandroff compactification ofX.

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 alimit point of${\displaystyle X}$ (i.e. not anisolated point of${\displaystyle X}$),${\displaystyle X}$ is the Alexandroff compactification of${\displaystyle X\setminus \{p\}}$.

LetX be any noncompactTychonoff 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.

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 calledone-point compactifications in this context. So one wants to determine all possible ways to give${\displaystyle X^{\ast }=X\cup \{\mathrm{\infty }\}}$ a compact topology such that${\displaystyle X}$ is dense in it and the subspace topology on${\displaystyle X}$ induced from${\displaystyle X^{\ast }}$ is the same as the original topology. The last compatibility condition on the topology automatically implies that${\displaystyle X}$ is dense in${\displaystyle X^{\ast }}$, because${\displaystyle X}$ is not compact, so it cannot be closed in a compact space.Also, it is a fact that the inclusion map${\displaystyle c:X\to X^{\ast }}$ is necessarily anopen embedding, that is,${\displaystyle X}$ must be open in${\displaystyle X^{\ast }}$ and the topology on${\displaystyle X^{\ast }}$ must contain every memberof${\displaystyle \tau }$.^[1]So the topology on${\displaystyle X^{\ast }}$ is determined by the neighbourhoods of${\displaystyle \mathrm{\infty }}$. Any neighborhood of${\displaystyle \mathrm{\infty }}$ is necessarily the complement in${\displaystyle X^{\ast }}$ of a closed compact subset of${\displaystyle X}$, as previously discussed.

The topologies on${\displaystyle X^{\ast }}$ 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${\displaystyle \mathrm{\infty }}$. This is the largest topology that makes${\displaystyle X^{\ast }}$ a one-point compactification of${\displaystyle X}$.
• Theopen extension topology. Here we add a single neighborhood of${\displaystyle \mathrm{\infty }}$, namely the whole space${\displaystyle X^{\ast }}$. This is the smallest topology that makes${\displaystyle X^{\ast }}$ a one-point compactification of${\displaystyle X}$.
• Any topology intermediate between the two topologies above. For neighborhoods of${\displaystyle \mathrm{\infty }}$ 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.

• The one-point compactification of the set of positive integers ishomeomorphic to the space consisting ofK = {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${\displaystyle f:{\mathbb{N}}^{\ast }\to X}$ given by${\displaystyle f(n)=a_n}$ for${\displaystyle n}$ in${\displaystyle \mathbb{N}}$ and${\displaystyle f(\mathrm{\infty })=a}$ is continuous. Here${\displaystyle \mathbb{N}}$ has thediscrete 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.

• The one-point compactification ofn-dimensional Euclidean spaceRⁿ is homeomorphic to then-sphereSⁿ. As above, the map can be given explicitly as ann-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 awedge of${\displaystyle n}$ circles.
• The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is theHawaiian 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 thequotient space.^[2]
• If${\displaystyle X}$ and${\displaystyle Y}$ are locally compact Hausdorff, then${\displaystyle (X\times Y{)}^{\ast }=X^{\ast }\wedge Y^{\ast }}$ where${\displaystyle \wedge }$ is thesmash 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 thewedge sum, and again, / denotes the quotient space.^[2]

The Alexandroff extension can be viewed as afunctor from thecategory of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps${\displaystyle c:X\to Y}$ and for which the morphisms from${\displaystyle c_1:X_1\to Y_1}$ to${\displaystyle c_2:X_2\to Y_2}$ are pairs of continuous maps${\displaystyle f_X:X_1\to X_2,f_Y:Y_1\to Y_2}$ such that${\displaystyle f_Y\circ c_1=c_2\circ f_X}$. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

• 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 - infinity added
• Normal space – Type of topological space
• 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 – Concept in topology
• Wallman compactification – A compactification of T₁ topological spaces

• 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

• General topology
• Compactification (mathematics)

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