Continuum (mathematics)

In mathematics , any set that has the power of real numbers is called a continuum .

Continua in general

One can show (e.g. with the ZF axioms , even without the axiom of choice ) that the following sets are all equally powerful:

1. ${\displaystyle \mathbb {R} }$, the set of all real numbers
2. ${\displaystyle \mathbb {C} }$, the set of all complex numbers
3. ${\displaystyle (0,1)}$, the set of all real numbers between 0 and 1
4. ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$, the set of all irrational numbers
5. ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$, the set of all real transcendent numbers
6. ${\displaystyle \mathbb {C} \setminus \mathbb {A} }$, the set of all complex transcendent numbers
7. ${\displaystyle {\mathcal {P}}({\mathbb {N} })}$, the set of all subsets of the natural numbers , i.e. the power set of ${\displaystyle \mathbb {N} }$
8. ${\displaystyle \left\{0,1\right\}^{\mathbb {N} }}$, the set of all functions with domain and target range {0,1} ${\displaystyle \mathbb {N} }$
9. ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$, the set of all sequences of natural numbers
10. ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$, the set of all sequences of real numbers
11. ${\displaystyle \{f\colon \mathbb {R} \to \mathbb {R} \mid f{\text{ stetig}}\}}$, the set of all continuous functions from to ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$
12. Every uncountable Polish space includes, with certain obvious interpretations, all previous examples, except for the sets of transcendent numbers, and also all at least one-dimensional manifolds .
13. ${\displaystyle {}^{*}\mathbb {R} }$, the set of all hyper-real numbers

The power of this set (or its cardinal number ) is usually called (Fraktur c, for c ontinuum), (see Beth function ) or (Aleph, the first letter of the Hebrew alphabet). Since this is the power set of concerns and the thickness of with is called, you also writes for . ${\displaystyle {\mathfrak {c}}}$ ${\displaystyle \beth _{1}}$ ${\displaystyle \aleph }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \aleph _{0}}$ ${\displaystyle 2^{\aleph _{0}}}$

It has been shown that many other structures that are examined in mathematics have the same thickness.

Continuum hypothesis

The assumption that all uncountable subsets of the real numbers are equal to the real numbers is called the continuum hypothesis . It is (with the usual axioms) neither refutable nor provable.

Continua in the topology

In topology , the concept of continuum is often more narrowly defined than in other sub-areas of mathematics . Here, a continuum is understood to be a coherent, compact Hausdorff space (term of continuum in the broader sense) .

Some authors additionally demand that a continuum must always satisfy the second axiom of countability , or even include the contiguous compact metric spaces under the concept of continuum (concept of continuum in the narrower sense) . Such a continuum in the narrower sense is therefore called (more accurately) a metric continuum (Engl. Metric continuum ). The metric continua provide many of the most important spaces in topology. Typical examples are:

1. Closed intervals of real numbers ${\displaystyle [a,b]\subset \mathbb {R} }$
2. Completed full spheres in -dimensional Euclidean space ${\displaystyle {\overline {B_{r}}}}$ ${\displaystyle n}$
3. The sphere in (n + 1) -dimensional Euclidean space ${\displaystyle n}$ ${\displaystyle S^{n}}$
4. Polygons
5. Jordan curves

The following sentence shows that the concept of the continuum that occurs in general in mathematics and that used in topology are not too far apart :

A metric continuum with more than one element has the power of the set of real numbers. ${\displaystyle {\mathfrak {c}}}$

Peano rooms

Peano spaces or Peano continuums are continua with special connected properties and are named after the Italian mathematician Giuseppe Peano . They too have different views on the question of the existence of a metric . After modern conception is a Peano-space (or Peanoraum . Engl Peano space or Peano continuum ) is a local coherent metric continuum with at least one element .

In his famous work Sur une courbe, qui remplit toute une aire plane in Volume 36 of the Mathematical Annals of 1890, Peano demonstrated that the unit interval can be mapped continuously to the square of the Euclidean plane . Upon further investigation of this surprising result, it was found that the Peano spaces the following characterization allow that today as a set of tap and Mazurkiewicz or as a set of tap-Mazurkiewicz-Sierpiński (after Stefan Mazurkiewicz , Hans Hahn and Wacław Sierpiński ) is known: ${\displaystyle I=[0,1]}$ ${\displaystyle I\times I=[0,1]\times [0,1]}$

A Hausdorff space is then and only then homeomorphic to a Peano space if a continuous mapping exists which is at the same time surjective . ${\displaystyle X\neq \emptyset }$ ${\displaystyle f\colon I\to X}$

In short, Peano spaces are the continuous images of the Peano curves except for homeomorphism .

literature

• Charles O. Christenson, William L. Voxman: Aspects of Topology . 2nd Edition. BCS Associates, Moscow, Idaho, USA 1998, ISBN 0-914351-08-7 .
• Lutz Führer : General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Stephen Willard: General Topology . Addison-Wesley, Reading, Massachusetts et al. 1970. MR0264581
• Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics. Second volume: differential calculus, infinite series, elements of differential geometry and function theory . 13th edition. S. Hirzel Verlag, Stuttgart 1967.
• Willi Rinow : Textbook of Topology . VEB Deutscher Verlag der Wissenschaften, 1975, ISBN 978-3-326-00433-4 .

References and footnotes

1. ↑ W. Rinow: Textbook of Topology . 1975, p. 223 ff .
2. ^ S. Willard: General Topology . 1970, p. 203 ff .
3. ↑ W. Rinow: Textbook of Topology . 1975, p. 223 .
4. ↑ L. Guide: General topology with applications . 1977, p. 125 ff .
5. ^ S. Willard: General Topology . 1970, p. 206 .
6. ↑ H. von Mangoldt, K. Knopp: Introduction to higher mathematics . tape 2 , 1967, p. 306 ff .
7. ↑ L. Guide: General topology with applications . 1977, p. 126 .
8. ↑ W. Rinow: Textbook of Topology . 1975, p. 223 .
9. ^ S. Willard: General Topology . 1970, p. 206 .
10. ↑ CO Christenson, WL Voxman: Aspects of Topology. 1998, p. 225 ff .
11. ^ S. Willard: General Topology . 1970, p. 219 ff .
12. ^ In W. Rinow: Textbook of Topology . 1975, p. 223 ff . every non-empty locally connected continuum with a countable base is called a peano space. Since such a device can always be measured according to Urysohn's metrization theorem , this makes no significant difference.
13. ↑ CO Christenson, WL Voxman: Aspects of Topology. 1998, p. 228 .
14. ↑ L. Guide: General topology with applications . 1977, p. 150, 154 .
15. ↑ W. Rinow: Textbook of Topology . 1975, p. 224 .
16. ^ S. Willard: General Topology . 1970, p. 221 .
17. ↑ H. von Mangoldt, K. Knopp: Introduction to higher mathematics . tape 2 , 1967, p. 406 ff .