Zero dimensional space

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

definition

A topological space is called zero-dimensional if it is zero-dimensional with regard to the Lebesgue cover dimension or with regard to the small or large inductive dimension , that is in formulas: ${\displaystyle X}$

• ${\displaystyle \mathrm {dim} (X)=0}$ (Lebesgue coverage dimension)
• ${\displaystyle \mathrm {Ind} (X)=0}$ (large inductive dimension)
• ${\displaystyle \mathrm {ind} (X)=0}$ (small inductive dimension)

Relationships

If it is not clear from the context which dimension is meant, it is said. In many cases this is not necessary because the following applies:

• For a normal space is considered , and it follows . ${\displaystyle X}$ ${\displaystyle \mathrm {dim} (X)=0\Leftrightarrow \mathrm {Ind} (X)=0}$ ${\displaystyle \mathrm {ind} (X)=0}$

In the important case of compact Hausdorff spaces , the following statements are equivalent: ${\displaystyle X}$

• ${\displaystyle \mathrm {dim} (X)=0}$.
• ${\displaystyle \mathrm {Ind} (X)=0}$.
• ${\displaystyle \mathrm {ind} (X)=0}$.
• ${\displaystyle X}$is totally incoherent .

In general, however, the relationships are not so simple, because there are totally incoherent, metrizable , separable spaces with and there are normal spaces with , and . ${\displaystyle \mathrm {dim} (X)>0}$ ${\displaystyle \mathrm {ind} (X)=0}$ ${\displaystyle \mathrm {dim} (X)=1}$ ${\displaystyle \mathrm {Ind} (X)=2}$

In any case, zero-dimensional Hausdorff spaces are spaces of whatever kind totally disconnected, the converse does not apply according to the above remarks, but it does for locally compact spaces .

Open-ended sets

Directly from the definitions it follows that a Hausdorff space has the small inductive dimension 0 if and only if it has a basis of open-closed sets . Therefore, one also finds this property as the definition of a zero-dimensional space, for example in. In the important case of compact Hausdorff rooms, this term also coincides with the above.

Individual evidence

1. ↑ Keiô Nagami: Dimension Theory. Academic Press, New York NY et al. 1970, ISBN 0-12-513650-1 ( Pure and Applied Mathematics 37), sentences 8-3.
2. ↑ Keiô Nagami: Dimension Theory. Academic Press, New York NY et al. 1970, ISBN 0-12-513650-1 ( Pure and Applied Mathematics 37), Clauses 8-4 and Clauses 8-6.
3. ↑ Keiô Nagami: Dimension Theory. Academic Press, New York NY et al. 1970, ISBN 0-12-513650-1 ( Pure and Applied Mathematics 37), sentences 9-12.
4. ↑ Keiô Nagami: Dimension Theory. Academic Press, New York NY et al. 1970, ISBN 0-12-513650-1 ( Pure and Applied Mathematics 37), chapter 19.
5. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 ( BI university pocket books 121), § 6, task 7.
6. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Dover Pubn Inc., New York 1995, ISBN 0-486-68735-X .