Zero-dimensional space

Zero-dimensional space is a term from the mathematical branch of topology. These are spaces of topological dimension 0, where it depends on the concept of dimension used.

Definition

A topological space is zero -dimensional, if it is with respect to the dimension or Lebesgue overlap with respect to the small or large dimension inductive zero -dimensional, that is, in the formulas:

• ( Lebesgue'sche covering dimension)
• (large inductive dimension)
• (small inductive dimension)

Relations

If the context is not clear which dimension is meant, so they say they do so. In many cases this is not necessary because it is:

• For a normal space, and it follows.

In the important case of compact Hausdorff spaces are the following statements are equivalent:

• .
• .
• .
• Is totally disconnected.

In general, however, are not so simple conditions before, because there are totally unrelated, metrizable, separable spaces with and there are normal rooms, and.

In any case, zero-dimensional Hausdorff spaces spaces of any kind totally disconnected, the converse is not true according to the above remarks, but rather for locally compact spaces.

Open - closed sets

Follows directly from the definitions that a Hausdorff space if and only the small inductive dimension 0, if it has a base of open - closed sets. Therefore, you can also find this property as the definition of a zero-dimensional space, such as in. In the important case of compact Hausdorff spaces also falls this term with the above together.