Editing Regular space (section)

== Examples and nonexamples ==
A [[zero-dimensional space]] with respect to the [[small inductive dimension]] has a [[Base (topology)|base]] consisting of [[clopen set]]s.
Every such space is regular.

As described above, any [[completely regular space]] is regular, and any T<sub>0</sub> space that is not [[Hausdorff space|Hausdorff]] (and hence not preregular) cannot be regular.
Most examples of regular and nonregular spaces studied in mathematics may be found in those two articles.
On the other hand, spaces that are regular but not completely regular, or preregular but not regular, are usually constructed only to provide [[counterexample]]s to conjectures, showing the boundaries of possible [[theorem]]s.
Of course, one can easily find regular spaces that are not T<sub>0</sub>, and thus not Hausdorff, such as an [[indiscrete space]], but these examples provide more insight on the [[Kolmogorov axiom|T<sub>0</sub> axiom]] than on regularity.  An example of a regular space that is not completely regular is the [[Tychonoff corkscrew]].

Most interesting spaces in mathematics that are regular also satisfy some stronger condition.
Thus, regular spaces are usually studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis.

There exist Hausdorff spaces that are not regular.  An example is the [[K-topology]] on the set <math>\R</math> of real numbers.  More generally, if <math>C</math> is a fixed nonclosed subset of <math>\R</math> with empty interior with respect to the usual Euclidean topology, one can construct a finer topology on <math>\R</math> by taking as a [[base (topology)|base]] the collection of all sets <math>U</math> and <math>U\setminus C</math> for <math>U</math> open in the usual topology.  That topology will be Hausdorff, but not regular.