Cardinality of the continuum

In mathematics it is called a lot, which has the cardinality of the real numbers, a continuum.

  • 2.1 Peano spaces

Continua in general

One can ( as with ZF axioms, even without the axiom of choice ) show that the following quantities are all equally powerful:

The cardinality of this set (or its cardinal number ) is usually called ( c fracture, for continuum ), (see Beth- function ) or ( Aleph, the first letter of the Hebrew alphabet). Since it is the power set of the cardinality and is denoted by with, to write this well.

It has been shown that many more structures that are investigated in mathematics, have the same cardinality.

Continuum hypothesis

The assumption that all uncountable subsets of the real numbers are equally powerful with the real numbers, is called the continuum hypothesis. It is (with the usual axioms ) neither refuted nor proven.

Continua in the topology

In the topology of the continuum term is often narrower than in other areas of mathematics. Here we mean by a continuum a continuous compact Hausdorff space ( continuum concept in a broader sense ).

Some authors call additionally that a continuum must always satisfy the second axiom of countability, or subsumed under the term continuum even just the contiguous compact metric spaces ( continuum term in the narrow sense ). Such a continuum in the narrow sense is called, therefore, (specifically ) a metric continuum (English metric continuum ). The metric continua supply many of the most important occurring in the topology of spaces. Typical examples would be:

The fact that in mathematics generally occurring and the continuum concept used in the topology are not too far apart, apparent from the following sentence:

Peano spaces

Peano continua are rooms with special connection properties, and are so named after the Italian mathematician Giuseppe Peano. Even with them, there are different views on the question of the existence of a metric. ( Peano space or Peano continuum engl or Peanoraum. ) Is a locally coherent metric continuum with at least one element of the modern view is a Peano space.

Peano pointed out in his famous paper Sur une courbe, qui toute une aire remplit plane after in Volume 36 of Mathematische Annalen in 1890 that the unit interval can be mapped to the square of the Euclidean plane in a continuous manner. Upon further investigation of this surprising result has been found that the Peano spaces admit the following characterization, which today as a set of Hahn and Mazurkiewicz or as Hahn- Mazurkiewicz - Sierpiński (after Stefan Mazurkiewicz, Hans Hahn and Waclaw Sierpiński ) known:

So In short, the Peano spaces up to homeomorphism the continuous images of Peano curves.