Locally compact groups are important because they have a natural measure called the Haar measure.

Locally compact groups have the stronger property of being normal.

Unlike the complex field, C is not locally compact.

For example, any locally compact preregular space is completely regular.

Every locally compact preregular space is, in fact, completely regular.

If the topological space is locally compact, these notions are equivalent.

Proper spaces are locally compact, but the converse is not true in general.

The rational numbers are an important example of a space which is not locally compact.

So let G be a locally compact group satisfying (3).

This has been extended in a number directions beyond the case that G is locally compact.