Prof. Girardi The Circle Group T Definition of Topological Group A
... metric dT ([θ1 ] , [θ2 ]) := inf k∈Z {|θ1 − θ2 + 2πk|}.
Let’s look at some nice properties of T. Consider the natural projection π : R T given by
π (θ) = [θ]. Then π is continuous since if dR (xn , x) → 0 then dT ([xn ] , [x]) → 0. Following directly
from the definition of the quotient topology is ...
Continuous function
In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.
