Alexandroff compactification

In the mathematical sub-area of topology , the Alexandroff compactification (also one-point compactification ) denotes the embedding of a non-compact topological space in a compact topological space by adding a single point. This compactification is named after the Russian mathematician Paul Alexandroff . In 1924, he and Heinrich Tietze recognized independently of one another that the construction of the Riemann number sphere derived from function theory can be generalized to this compactification. It is uniquely determined for locally compact Hausdorff spaces except for homeomorphism .

definition

Be a topological space and an element that does not come from . In addition, let the set with the topology ${\displaystyle (X,{\mathcal {T}})}$ ${\displaystyle \infty }$ ${\displaystyle X}$ ${\displaystyle X^{*}:=X\cup \{\infty \}}$

${\displaystyle {\mathcal {T}}^{*}:={\mathcal {T}}\cup \{X^{*}\setminus A\mid A\subseteq X,A{\text{ ist abgeschlossen und kompakt in }}(X,{\mathcal {T}})\}}$

fitted. Then there is a compact space that contains as an open subspace . The compactification is through canonical injection ${\displaystyle (X^{*},{\mathcal {T}}^{*})}$ ${\displaystyle (X,{\mathcal {T}})=(X,{\mathcal {T}}_{X^{*}}^{*})}$

${\displaystyle \iota \colon X\to X^{*},\quad \iota (x):=x}$

given. The Alexandroff compactification of is often called instead of the space , provided that it is a dense subset of . ${\displaystyle \iota }$ ${\displaystyle (X^{*},{\mathcal {T}}^{*})}$ ${\displaystyle (X,{\mathcal {T}})}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$

The point is sometimes referred to as infinitely far . ${\displaystyle \infty }$

properties

The above construction exists for any topological space . However, it actually only provides a compactification for spaces that are not yet compact themselves: If the topological space formed according to the previous definition, then the one-point set is open if it is assumed to be compact. In this case it is not tightly in and consequently the injection does not provide any compacting. ${\displaystyle (X,{\mathcal {T}})}$ ${\displaystyle (X^{*},{\mathcal {T}}^{*})}$ ${\displaystyle \{\infty \}}$ ${\displaystyle X}$ ${\displaystyle X=\iota (X)}$ ${\displaystyle X^{*}}$ ${\displaystyle \iota }$

It is advantageous if a compaction preserves the separation properties of a topological space. Thus the Alexandroff compactification z. B. the T ₁ axiom . However, the Hausdorff property is only retained if it is also assumed to be locally compact . But then the Alexandroff compactification is clearly defined in the following sense: ${\displaystyle (X,{\mathcal {T}})}$

Let and be compact Hausdorff spaces and also a (locally compact) subspace of the same, where and apply, so are and homeomorphic . ${\displaystyle (X_{1},{\mathcal {T}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {T}}_{2})}$ ${\displaystyle (X,{\mathcal {T}})}$ ${\displaystyle X_{1}\setminus X=\{\infty _{1}\}}$ ${\displaystyle X_{2}\setminus X=\{\infty _{2}\}}$ ${\displaystyle (X_{1},{\mathcal {T}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {T}}_{2})}$

Examples

• The projective extension of the real numbers is, together with the correspondingly extended topology, an Alexandroff compactification of the locally compact space of the real numbers with Euclidean topology . It is homeomorphic to the circular line . ${\displaystyle \mathbb {R^{*}} :=\mathbb {R} \cup \{\infty \}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {S} ^{1}}$
• The Riemann number sphere is, similar to the previous example, an Alexandroff compactification, through which one obtains a homeomorphy to the sphere . ${\displaystyle \mathbb {C^{*}} :=\mathbb {C} \cup \{\infty \}}$ ${\displaystyle \mathbb {S} ^{2}}$
• More generally, the Alexandroff compactification of with Euclidean topology is homeomorphic to the unitary sphere . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {S} ^{n}}$
• If a not compact but locally compact Hausdorff space, then the Banach algebra of the continuous functions on its Alexandroff compactification is isometrically isomorphic to the algebra of the continuous functions , which vanish in infinity , after the adjunct of a one element . ${\displaystyle X}$ ${\displaystyle C(X^{*})}$ ${\displaystyle {\widetilde {C_{0}(X)}}}$ ${\displaystyle X}$

Multipoint compactifications

If one embeds a topological space in a compact space that contains a finite number of points more, one speaks of a multipoint compactification or, in the case of additional points, of a -point compactification. This idea can be further generalized to countable compactifications. ${\displaystyle N}$ ${\displaystyle N}$

definition

Be and a topological space and a compact space. A compactification ${\displaystyle N\in \mathbb {N} ,N\geq 1}$ ${\displaystyle (X,\tau )}$ ${\displaystyle (Y,\sigma )}$

${\displaystyle \iota \colon X\to Y}$

means -point compactification of , if ${\displaystyle N}$ ${\displaystyle X}$

${\displaystyle |Y\setminus X|=N}$

applies.

properties

The following two statements are equivalent for topological spaces : ${\displaystyle (X,\tau )}$

• The room has a -point compactification with Hausdorff properties. ${\displaystyle X}$ ${\displaystyle N}$ ${\displaystyle (Y,\sigma )}$
• The space is a locally compact Hausdorff space and there exists a -element family of non-empty pairwise disjoint subsets , so that on the one hand ${\displaystyle X}$ ${\displaystyle N}$ ${\displaystyle (V_{i})_{i=1,\dots ,n}}$ ${\displaystyle V_{1},\dots ,V_{N}\in \tau }$

${\displaystyle K:=X\setminus (V_{1}\cup \cdots \cup V_{N})}$

is compact and on the other hand the quantity for each ${\displaystyle k=1,\dots ,N}$

${\displaystyle X\setminus (V_{1}\cup \cdots V_{k-1}\cup V_{k+1}\cup \cdots V_{N})=K\cup V_{k}}$

is no longer compact.

If there is a -point compactification, then in particular there is also a -point compactification for all . ${\displaystyle X}$ ${\displaystyle N}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle M<N}$

A -element family in the sense of the above characterization is also called a -star. Every star gives rise to a point compactification. An equivalence relation can be defined on the set of all stars as follows : ${\displaystyle N}$ ${\displaystyle (V_{i})_{i=1,\dots ,N}}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N}$

Two stars and are equivalent if ${\displaystyle N}$ ${\displaystyle (V_{i})_{i=1,\dots ,N}}$ ${\displaystyle (W_{i})_{i=1,\dots ,N}}$

${\displaystyle {\big (}X\setminus (V_{1}\cup \cdots \cup V_{N}){\big )}\cap {\big (}X\setminus W_{k}{\big )}}$

is compact, for everyone . ${\displaystyle k=1,\dots ,N}$

There is a 1-to-1 relationship between equivalence classes of -star and -point compactifications. ${\displaystyle N}$ ${\displaystyle N}$

Examples

• The affine extension of the real numbers is just the two-point compactification of . The real numbers only have -point compactifications for . ${\displaystyle [-\infty ,\infty ]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle N\leq 2}$

• The complex numbers and, more generally, the Euclidean with have no -point compactification for . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n>1}$ ${\displaystyle N}$ ${\displaystyle N>1}$

• For every natural number there is a topological space which has a -point compactification but no -point compactification for : ${\displaystyle N>0}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle M>N}$

Just look at the rays

${\displaystyle L_{n}:={\big \{}(x,y)\in \mathbb {R} ^{2}\mid y=x\cdot n^{-1},\;x\geq 0{\big \}},\quad n=1,\dots ,N}$,

and their union

${\displaystyle S_{N}:=\bigcup _{n=1}^{N}L_{n}}$

as a topological space with subspace topology. For true even ${\displaystyle K=\{(0,0)\}}$

${\displaystyle S_{N}\setminus K=\bigcup _{n=1}^{N}{\big (}L_{n}\setminus K{\big )}}$

and is compact for no . ${\displaystyle K\cup (L_{k}\setminus K)}$ ${\displaystyle k=1,\dots ,N}$

See also

• Stone-Čech compacting
• Wallman compactification

literature

• René Bartsch: General Topology . 2nd Edition. De Gruyter , Berlin 2015, ISBN 978-3-11-040618-4 .
• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer Verlag , Berlin 2001, ISBN 3-540-67790-9 .

Web links

• Karsten Evers: Set theoretical topology. P. 83 , accessed on December 26, 2016 (contains, among other things, a sentence about the existence of T2 multipoint compactifications). 

Individual evidence

1. ↑ Paul Alexandroff: On the metrization of the small compact topological spaces . In: Springer-Verlag (ed.): Mathematische Annalen . tape 92 , no. 3-4 , 1924, pp. 294-301 , doi : 10.1007 / BF01448011 .
2. ^ Heinrich Tietze: Contributions to the general topology. II. About the introduction of improper elements . In: Springer-Verlag (ed.): Mathematische Annalen . tape 91 , no. 3-4 , 1924, pp. 210-224 , doi : 10.1007 / BF01556079 .
3. ↑ René Bartsch: General Topology . de Gruyter, 2015, ISBN 978-3-110-40618-4 , p. 183 ( limited preview in Google book search).
4. ^ Boto von Querenburg: Set theoretical topology. 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 , p. 110 ( limited preview in the Google book search).
5. ↑ Lynn Arthur Steen: Counterexamples in Topology . Courier Corporation, 1995, p. 63, ISBN 978-0-486-68735-3 .
6. ↑ James R. Munkres: Topology. Prentice Hall, 2000, p. 185, ISBN 978-0-131-78449-9 .
7. ^ Lutz Führer: General topology with applications. Springer-Verlag, 2013, ISBN 978-3-322-84064-6 , p. 108 ( limited preview in the Google book search).
8. ^ Eberhard Kaniuth: A Course in Commutative Banach Algebras . Springer Science & Business Media, 2008, ISBN 978-0-387-72476-8 ( limited preview in Google book search).
9. ↑ ^{a b c} K. D. Magill, Jr .: N-Point Compactifications . In: Mathematical Association of America (Ed.): The American Mathematical Monthly . Vol. 72, No. 10 , 1965, p. 1075-1081 , doi : 10.2307 / 2315952 .
10. ^ KD Magill, Jr .: Countable Compactifications . In: Canadian Mathematical Society (Ed.): Canadian Journal of Mathematics . Vol. 18, 1966, pp. 616-620 , doi : 10.4153 / CJM-1966-060-6 .
11. ↑ KG Binmore: The Foundations of Topological Analysis: A Straightforward Introduction. Cambridge University Press, 1980, ISBN 978-0-521-29930-5 , p. 154 ( limited preview in Google Book Search).