Topological dimension

One gives the traditional definition here, by recurrence, of the topological dimension of a space métrisable at countable base E . If E is empty, its dimension is worth -1 per convention; if not one poses:

0) space E is of dimension 0 if its topology admits a base of at the same time open and closed parts ( clopen in English), that is to say still a base of parts at empty border (or of dimension -1). It is also said that E is completely discontinuous.
1) space E is with more dimension 1 if its topology admits a base of open to border of dimension to more the 0.
$\ vdots$
N) space E is of dimension to more N if its topology admits a base of open to border of dimension to more N -1.

Finally nonempty space E is known as of dimension N if it is of dimension to more N but is not dimension with the more n-1 , and of infinite dimension if there does not exist N such as there is of dimension to more N .

The Ensemble of Cantor (or the Espace of Cantor which is homeomorphic for him) is a compact Espace of dimension 0; space N^N is the paragon of the Polish spaces of dimension 0. A rectifiable Arc of Jordan in R ^N is of dimension 1, a portion of regular surface is of dimension 2, etc As it should be the dimension of all open not vacuum of R ^N is N .

Dimension introduced above, with whole value, is a topological concept whereas the concept of Dimension of Hausdorff, with actual value, is metric, and strongly depends on the distance used. There exists however a beautiful relation between the two when E is a compact Espace métrisable:

the topological dimension of E is the minimum dimensions of Hausdorff of E for all the distances on E compatible with its topology.