Topological and Analytic Structure
The circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on C×, the circle group has the structure of a topological group. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of C× (itself regarded as a topological group).
One can say even more. The circle is a 1-dimensional real manifold and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to Tn.
Read more about this topic: Circle Group
Famous quotes containing the words analytic and/or structure:
““You, that have not lived in thought but deed,
Can have the purity of a natural force,
But I, whose virtues are the definitions
Of the analytic mind, can neither close
The eye of the mind nor keep my tongue from speech.””
—William Butler Yeats (1865–1939)
“I’m a Sunday School teacher, and I’ve always known that the structure of law is founded on the Christian ethic that you shall love the Lord your God and your neighbor as yourself—a very high and perfect standard. We all know the fallibility of man, and the contentions in society, as described by Reinhold Niebuhr and many others, don’t permit us to achieve perfection.”
—Jimmy Carter (James Earl Carter, Jr.)