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In [[mathematics|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 space|compact]]. It is named for the Russian mathematician [[Pavel Alexandrov]].
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More precisely, let ''X'' be a topological space. Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an [[open mapping|open]] [[embedding (topology)|embedding]] ''c''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞. The map ''c'' is a Hausdorff [[compactification (mathematics)|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 [[Tychonoff space]], a much larger class of spaces.
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== 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 <math>S^{-1}: \mathbb{R}^2 \hookrightarrow S^2</math> is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point <math>\infty = (0,0,1)</math>. Under the stereographic projection latitudinal circles <math>z = c</math> get mapped to planar circles <math>r = \sqrt{\frac{1+c}{1-c}}</math>.  It follows that the deleted neighborhood basis of <math>(1,0,0)</math> given by the punctured spherical caps <math>c \leq z < 1</math> corresponds to the complements of closed planar disks <math>r \geq \sqrt{\frac{1+c}{1-c}}</math>.  More qualitatively, a neighborhood basis at <math>\infty</math> is furnished by the sets <math>S^{-1}(\mathbb{R}^2
\setminus K) \cup \{ \infty \}</math> as ''K'' ranges through the compact subsets of <math>\mathbb{R}^2</math>. This example already contains the key concepts of the general case.
 
== Motivation ==
 
Let <math>c: X \hookrightarrow Y</math> be an embedding from a topological space ''X'' to a compact Hausdorff topological space ''Y'', with dense image and one-point remainder <math>\{ \infty \} = Y \setminus c(X)</math>. 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 <math>\infty</math> must be all sets obtained by adjoining <math>\infty</math> to the image under ''c'' of a subset of ''X'' with compact complement.
 
== The Alexandroff extension ==
 
Let ''X'' be any topological space, and let <math>\infty</math> be any object which is not already an element of ''X''. (In terms of formal set theory one could take, for example, <math>\infty</math> to be ''X'' itself, but it is not really necessary or helpful to be so specific.)  Put <math>X^* = X \cup \{\infty \}</math>, and topologize <math>X^*</math> by taking as open sets all the open subsets ''U'' of ''X'' together with all subsets ''V'' which contain <math>\infty</math> and such that <math>X \setminus V</math> is closed and compact, {{harv|Kelley|1975|p=150}}.
 
The inclusion map <math>c: X \rightarrow X^*</math> is called the '''Alexandroff extension''' of ''X'' (Willard, 19A).
 
The above properties all follow easily from the above discussion:
* The map ''c'' is continuous and open: it embeds ''X'' as an open subset of <math>X^*</math>.
* The space <math>X^*</math> is compact.
* The image ''c''(''X'') is dense in <math>X^*</math>, if ''X'' is noncompact.
* The space <math>X^*</math> is [[Hausdorff space|Hausdorff]] if and only if ''X'' is Hausdorff and [[locally compact]].
 
== The one-point compactification ==
 
In particular, the Alexandroff extension <math>c: X \rightarrow X^*</math> 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 <math>\mathcal{C}(X)</math> 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 ==
 
* The one-point compactification of the set of positive integers is [[Homeomorphism|homeomorphic]] to the space consisting of ''K'' = {0} U {''1/n'' | ''n'' is a positive integer.} with the order topology.
 
* The one-point compactification of ''n''-dimensional Euclidean space ''R''<sup>n</sup> is homeomorphic to the ''n''-sphere ''S''<sup>n</sup>.  As above, the map can be given explicitly as an ''n''-dimensional inverse stereographic projection.
 
* 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 <math>\kappa</math> copies of the interval ''(0,1)'' is a [[Bouquet of circles|wedge of <math>\kappa</math> circles]].
 
* The Alexandroff extension can be viewed as a functor from the category of topological spaces to the category whose objects are continuous maps <math>c: X \rightarrow Y</math> and for which the morphisms from <math>c_1: X_1 \rightarrow Y_1</math> to <math>c_2: X_2 \rightarrow Y_2</math> are pairs of continuous maps <math>f_X: X_1 \rightarrow X_2, \ f_Y:
Y_1 \rightarrow Y_2</math> such that <math>f_Y \circ c_1 = c_2 \circ f_X</math>.  In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
 
==See also==
*[[Wallman compactification]]
*[[End (topology)]]
*[[Riemann sphere]]
*[[Normal space]]
*[[Stereographic projection]]
 
==References==
 
*{{citation|author=[[Pavel Alexandroff|P.S. Alexandroff]]|journal=  Math. Ann. |volume= 92|issue=3-4  |year=1924|pages= 294–301|title= Über die Metrisation der im Kleinen kompakten topologischen Räume|doi=10.1007/BF01448011}}
*{{citation|
author=[[Ronald Brown (mathematician)|Ronald Brown]]
| journal= J. London Math Soc.
|volume = (2) 7
| year= 1973
| pages= 515–522
|title= Sequentially proper maps and a sequential compactification}}
 
*{{Citation | last1=Engelking | first1=Ryszard | authorlink = Ryszard Engelking | title=General Topology | publisher=[[Helderman Verlag Berlin]] | isbn=978-0-201-08707-9 | mr=1039321 | year=1989}}
*{{springer|id=A/a011260|first=V.V. |last=Fedorchuk|title=Aleksandrov compactification}}
* {{Citation | last1=Kelley | first1=John L. | author1-link=John L. Kelley | title=General Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90125-1 | mr=0370454  | year=1975}}
* {{Citation
| author = [[James Munkres]]
| year = 1999
| title = Topology
| edition = 2nd
| publisher = [[Prentice Hall]]
| isbn = 0-13-181629-2
}}
*{{Citation | last1=Willard | first1=Stephen | title=General Topology | publisher=[[Addison-Wesley]] | isbn=3-88538-006-4 | mr=0264581 | year=1970}}
 
[[Category:General topology]]
[[Category:Compactification]]

Latest revision as of 22:00, 5 January 2015

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