• Source: 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




S

−
1


:


R


2


↪

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



∞
=
(
0
,
0
,
1
)


{\displaystyle \infty =(0,0,1)}

. Under the stereographic projection latitudinal circles



z
=
c


{\displaystyle z=c}

get mapped to planar circles



r
=


(
1
+
c
)

/

(
1
−
c
)




{\textstyle r={\sqrt {(1+c)/(1-c)}}}

. It follows that the deleted neighborhood basis of



(
0
,
0
,
1
)


{\displaystyle (0,0,1)}

given by the punctured spherical caps



c
≤
z
<
1


{\displaystyle c\leq z<1}

corresponds to the complements of closed planar disks



r
≥


(
1
+
c
)

/

(
1
−
c
)




{\textstyle r\geq {\sqrt {(1+c)/(1-c)}}}

. More qualitatively, a neighborhood basis at



∞


{\displaystyle \infty }

is furnished by the sets




S

−
1


(


R


2


∖
K
)
∪
{
∞
}


{\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}}

as K ranges through the compact subsets of





R


2




{\displaystyle \mathbb {R} ^{2}}

. This example already contains the key concepts of the general case.




Motivation


Let



c
:
X
↪
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



{
∞
}
=
Y
∖
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 }

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



X


{\displaystyle X}

be a topological space. Put




X

∗


=
X
∪
{
∞
}
,


{\displaystyle X^{*}=X\cup \{\infty \},}

and topologize




X

∗




{\displaystyle X^{*}}

by taking as open sets all the open sets in X together with all sets of the form



V
=
(
X
∖
C
)
∪
{
∞
}


{\displaystyle V=(X\setminus C)\cup \{\infty \}}

where C is closed and compact in X. Here,



X
∖
C


{\displaystyle X\setminus C}

denotes the complement of



C


{\displaystyle C}

in



X
.


{\displaystyle X.}

Note that



V


{\displaystyle V}

is an open neighborhood of



∞
,


{\displaystyle \infty ,}

and thus any open cover of



{
∞
}


{\displaystyle \{\infty \}}

will contain all except a compact subset



C


{\displaystyle C}

of




X

∗


,


{\displaystyle X^{*},}

implying that




X

∗




{\displaystyle X^{*}}

is compact (Kelley 1975, p. 150).
The space




X

∗




{\displaystyle X^{*}}

is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map



c
:
X
→

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




X

∗




{\displaystyle X^{*}}

.
The space




X

∗




{\displaystyle X^{*}}

is compact.
The image c(X) is dense in




X

∗




{\displaystyle X^{*}}

, if X is noncompact.
The space




X

∗




{\displaystyle X^{*}}

is Hausdorff if and only if X is Hausdorff and locally compact.
The space




X

∗




{\displaystyle X^{*}}

is T1 if and only if X is T1.




The one-point compactification


In particular, the Alexandroff extension



c
:
X
→

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



X


{\displaystyle X}

is a compact Hausdorff space and



p


{\displaystyle p}

is a limit point of



X


{\displaystyle X}

(i.e. not an isolated point of



X


{\displaystyle X}

),



X


{\displaystyle X}

is the Alexandroff compactification of



X
∖
{
p
}


{\displaystyle X\setminus \{p\}}

.
Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set





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.




Non-Hausdorff one-point compactifications


Let



(
X
,
τ
)


{\displaystyle (X,\tau )}

be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of



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




X

∗


=
X
∪
{
∞
}


{\displaystyle X^{*}=X\cup \{\infty \}}

a compact topology such that



X


{\displaystyle X}

is dense in it and the subspace topology on



X


{\displaystyle X}

induced from




X

∗




{\displaystyle X^{*}}

is the same as the original topology. The last compatibility condition on the topology automatically implies that



X


{\displaystyle X}

is dense in




X

∗




{\displaystyle X^{*}}

, because



X


{\displaystyle X}

is not compact, so it cannot be closed in a compact space.
Also, it is a fact that the inclusion map



c
:
X
→

X

∗




{\displaystyle c:X\to X^{*}}

is necessarily an open embedding, that is,



X


{\displaystyle X}

must be open in




X

∗




{\displaystyle X^{*}}

and the topology on




X

∗




{\displaystyle X^{*}}

must contain every member
of



τ


{\displaystyle \tau }

.
So the topology on




X

∗




{\displaystyle X^{*}}

is determined by the neighbourhoods of



∞


{\displaystyle \infty }

. Any neighborhood of



∞


{\displaystyle \infty }

is necessarily the complement in




X

∗




{\displaystyle X^{*}}

of a closed compact subset of



X


{\displaystyle X}

, as previously discussed.
The topologies on




X

∗




{\displaystyle X^{*}}

that make it a compactification of



X


{\displaystyle X}

are as follows:

The Alexandroff extension of



X


{\displaystyle X}

defined above. Here we take the complements of all closed compact subsets of



X


{\displaystyle X}

as neighborhoods of



∞


{\displaystyle \infty }

. This is the largest topology that makes




X

∗




{\displaystyle X^{*}}

a one-point compactification of



X


{\displaystyle X}

.
The open extension topology. Here we add a single neighborhood of



∞


{\displaystyle \infty }

, namely the whole space




X

∗




{\displaystyle X^{*}}

. This is the smallest topology that makes




X

∗




{\displaystyle X^{*}}

a one-point compactification of



X


{\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



X


{\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



{

a

n


}


{\displaystyle \{a_{n}\}}

in a topological space



X


{\displaystyle X}

converges to a point



a


{\displaystyle a}

in



X


{\displaystyle X}

, if and only if the map



f
:


N


∗


→
X


{\displaystyle f\colon \mathbb {N} ^{*}\to X}

given by



f
(
n
)
=

a

n




{\displaystyle f(n)=a_{n}}

for



n


{\displaystyle n}

in




N



{\displaystyle \mathbb {N} }

and



f
(
∞
)
=
a


{\displaystyle f(\infty )=a}

is continuous. Here




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.




= 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



κ


{\displaystyle \kappa }

copies of the half-closed interval [0,1), that is, of



[
0
,
1

)

κ




{\displaystyle [0,1)^{\kappa }}

, is (homeomorphic to)



[
0
,
1

]

κ




{\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



n


{\displaystyle n}

of copies of the interval (0,1) is a wedge of



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



X


{\displaystyle X}

compact Hausdorff and



C


{\displaystyle C}

any closed subset of



X


{\displaystyle X}

, the one-point compactification of



X
∖
C


{\displaystyle X\setminus C}

is



X

/

C


{\displaystyle X/C}

, where the forward slash denotes the quotient space.
If



X


{\displaystyle X}

and



Y


{\displaystyle Y}

are locally compact Hausdorff, then



(
X
×
Y

)

∗


=

X

∗


∧

Y

∗




{\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}}

where



∧


{\displaystyle \wedge }

is the smash product. Recall that the definition of the smash product:



A
∧
B
=
(
A
×
B
)

/

(
A
∨
B
)


{\displaystyle A\wedge B=(A\times B)/(A\vee B)}

where



A
∨
B


{\displaystyle A\vee B}

is the wedge sum, and again, / denotes the quotient space.




= 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
:
X
→
Y


{\displaystyle c\colon X\rightarrow Y}

and for which the morphisms from




c

1


:

X

1


→

Y

1




{\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}}

to




c

2


:

X

2


→

Y

2




{\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}}

are pairs of continuous maps




f

X


:

X

1


→

X

2


,


f

Y


:

Y

1


→

Y

2




{\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}}

such that




f

Y


∘

c

1


=

c

2


∘

f

X




{\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 - 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




Notes






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

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