Topological group

Summary

A topological group is a group with a topology in which multiplication and inversion are continuous. It is the basic object where algebra and topology work together. It is the setting of abstract harmonic analysis and of the theory of Lie groups. Translations by group elements are homeomorphisms, so the topology near any point looks like the topology near the identity. The real line under addition, the circle group, matrix groups and the $p$-adic integers are all topological groups.