Examples and Nonexamples
A zero-dimensional space with respect to the small inductive dimension has a base consisting of clopen sets. Every such space is regular.
As described above, any completely regular space is regular, and any T0 space that is not 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 counterexamples to conjectures, showing the boundaries of possible theorems. Of course, one can easily find regular spaces that are not T0, and thus not Hausdorff, such as an indiscrete space, but these examples provide more insight on the T0 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 set R with the topology generated by sets of the form U — C, where U is an open set in the usual sense, and C is any countable subset of U.
Read more about this topic: Regular Space
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“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)