zero dimensional
Definition 1.
[1, 2] Suppose $X$ is a topological space^{}. If $X$ has a basis consising of clopen sets, then $X$ is said to be .
Examples of zero-dimensional spaces are: the set $\mathbb{Q}$ of rational numbers^{} (with subspace topology induced from the usual metric topology on $\mathbb{R}$, the set of real numbers), the Cantor space, as well as the Sorgenfrey line.
The concepts of zero-dimentionality and total disconnectedness are closely related. Indeed, every zero-dimentional ${T}_{1}$ space (http://planetmath.org/T1Space) is totally disconnected. Furthermore, if a topological space is locally compact and Hausdorff^{}, then the notions of zero-dimentionality and total disconnectedness are equivalent^{}.
References
- 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
- 2 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title | zero dimensional |
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Canonical name | ZeroDimensional |
Date of creation | 2013-03-22 14:41:05 |
Last modified on | 2013-03-22 14:41:05 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 9 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | zero-dimensional |
Related topic | SeparationAxioms |