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).

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 ${\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}}${\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 ${\displaystyle \infty =(0,0,1)}${\displaystyle \infty =(0,0,1)}. Under the stereographic projection latitudinal circles ${\displaystyle z=c}${\displaystyle z=c} get mapped to planar circles ${\textstyle r={\sqrt {(1+c)/(1-c)}}}${\textstyle r={\sqrt {(1+c)/(1-c)}}}. It follows that the deleted neighborhood basis of ${\displaystyle (0,0,1)}${\displaystyle (0,0,1)} given by the punctured spherical caps ${\displaystyle c\leq z<1}${\displaystyle c\leq z<1} corresponds to the complements of closed planar disks ${\textstyle r\geq {\sqrt {(1+c)/(1-c)}}}${\textstyle r\geq {\sqrt {(1+c)/(1-c)}}}. More qualitatively, a neighborhood basis at ${\displaystyle \infty }${\displaystyle \infty } is furnished by the sets ${\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}}${\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}} as K ranges through the compact subsets of ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}}. This example already contains the key concepts of the general case.

Let ${\displaystyle c:X\hookrightarrow Y}${\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 ${\displaystyle \{\infty \}=Y\setminus c(X)}${\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 ${\displaystyle \infty }${\displaystyle \infty } must be all sets obtained by adjoining ${\displaystyle \infty }${\displaystyle \infty } to the image under c of a subset of X with compact complement.

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

The space ${\displaystyle X^{*}}${\displaystyle X^{*}} is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map ${\displaystyle c:X\to X^{*}.}${\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 ${\displaystyle X^{*}}${\displaystyle X^{*}}.
• The space ${\displaystyle X^{*}}${\displaystyle X^{*}} is compact.
• The image c(X) is dense in ${\displaystyle X^{*}}${\displaystyle X^{*}}, if X is noncompact.
• The space ${\displaystyle X^{*}}${\displaystyle X^{*}} is Hausdorff if and only if X is Hausdorff and locally compact.
• The space ${\displaystyle X^{*}}${\displaystyle X^{*}} is T₁ if and only if X is T₁.

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

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

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

• The Alexandroff extension of ${\displaystyle X}${\displaystyle X} defined above. Here we take the complements of all closed compact subsets of ${\displaystyle X}${\displaystyle X} as neighborhoods of ${\displaystyle \infty }${\displaystyle \infty }. This is the largest topology that makes ${\displaystyle X^{*}}${\displaystyle X^{*}} a one-point compactification of ${\displaystyle X}${\displaystyle X}.
• The open extension topology. Here we add a single neighborhood of ${\displaystyle \infty }${\displaystyle \infty }, namely the whole space ${\displaystyle X^{*}}${\displaystyle X^{*}}. This is the smallest topology that makes ${\displaystyle X^{*}}${\displaystyle X^{*}} a one-point compactification of ${\displaystyle X}${\displaystyle X}.
• Any topology intermediate between the two topologies above. For neighborhoods of ${\displaystyle \infty }${\displaystyle \infty } one has to pick a suitable subfamily of the complements of all closed compact subsets of ${\displaystyle X}${\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 is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer} with the order topology.
• A sequence ${\displaystyle \{a_{n}\}}${\displaystyle \{a_{n}\}} in a topological space ${\displaystyle X}${\displaystyle X} converges to a point ${\displaystyle a}${\displaystyle a} in ${\displaystyle X}${\displaystyle X}, if and only if the map ${\displaystyle f\colon \mathbb {N} ^{*}\to X}${\displaystyle f\colon \mathbb {N} ^{*}\to X} given by ${\displaystyle f(n)=a_{n}}${\displaystyle f(n)=a_{n}} for ${\displaystyle n}${\displaystyle n} in ${\displaystyle \mathbb {N} }${\displaystyle \mathbb {N} } and ${\displaystyle f(\infty )=a}${\displaystyle f(\infty )=a} is continuous. Here ${\displaystyle \mathbb {N} }${\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.

• 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 ${\displaystyle \kappa }${\displaystyle \kappa } copies of the half-closed interval [0,1), that is, of ${\displaystyle [0,1)^{\kappa }}${\displaystyle [0,1)^{\kappa }}, is (homeomorphic to) ${\displaystyle [0,1]^{\kappa }}${\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 ${\displaystyle n}${\displaystyle n} of copies of the interval (0,1) is a wedge of ${\displaystyle n}${\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 ${\displaystyle X}${\displaystyle X} compact Hausdorff and ${\displaystyle C}${\displaystyle C} any closed subset of ${\displaystyle X}${\displaystyle X}, the one-point compactification of ${\displaystyle X\setminus C}${\displaystyle X\setminus C} is ${\displaystyle X/C}${\displaystyle X/C}, where the forward slash denotes the quotient space.^[2]
• If ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} are locally compact Hausdorff, then ${\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}}${\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}} where ${\displaystyle \wedge }${\displaystyle \wedge } is the smash product. Recall that the definition of the smash product:${\displaystyle A\wedge B=(A\times B)/(A\vee B)}${\displaystyle A\wedge B=(A\times B)/(A\vee B)} where ${\displaystyle A\vee B}${\displaystyle A\vee B} is the wedge sum, and again, / denotes the quotient space.^[2]

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 ${\displaystyle c\colon X\rightarrow Y}${\displaystyle c\colon X\rightarrow Y} and for which the morphisms from ${\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}}${\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}} to ${\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}}${\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}} are pairs of continuous maps ${\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}}${\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}} such that ${\displaystyle f_{Y}\circ c_{1}=c_{2}\circ f_{X}}${\displaystyle f_{Y}\circ c_{1}=c_{2}\circ f_{X}}. In particular, homeomorphic spaces have isomorphic Alexandroff extensions. The latter is the arrow category of topological spaces, often constructed as category of functors from interval category ${\textstyle \operatorname {Arr} (\mathrm {Top} )=\operatorname {Func} (\mathrm {I} ,\mathrm {Top} )}${\textstyle \operatorname {Arr} (\mathrm {Top} )=\operatorname {Func} (\mathrm {I} ,\mathrm {Top} )}, where interval category is the category with 2 objects connected by single arrow.

• Bohr compactification
• Compact space – Type of mathematical space
• Compactification (mathematics) – Embedding a topological space into a compact space as a dense subset
• End (topology)
• 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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