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
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