Space of Cantor
One calls space of Cantor space produced . It is a compact Espace métrisable at countable base (in fact, for a compact space, being métrisable or being at countable base is equivalent properties, completely discontinuous (one says also dimension 0), who with the following universal property:
- All space métrisable at countable base completely discontinuous is homeomorphic with a subspace of K.
That provides in particular a convenient means for compactifier completely discontinuous spaces métrisables with countable base. One from of deduced that all measurable Espace dénombrablement generated and separated is isomorphous with part of K provided with the tribe induced by the Tribu borélienne with K .
The space of Cantor is homeomorphic with the Ensemble of Cantor, but it is equipped naturally with a ultrametric distance similar to that on NN of which one will find a description in the article “swell”. It is also, in Probabilité, the canonical space on which one builds the Jeu of pile or face.
The space of Cantor K with the power of the continuous one, and it is shown for example that the boréliens of a compact space métrisable have the power of continuous as soon as they are not-countable by proving that they contain a homeomorphic subspace with K .
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