Convergence and Continuity

Abstract

The idea of a sequence converging to a limit is defined rigorously in a general metric space. Examples are taken from real analysis, and it is shown that the sequences of a subset $A$ can only converge to limits in $\bar{A}$. As convergence is the most important structure of metric spaces, the functions that preserve convergence take on a paramount importance: they are the continuous functions. Three equivalent formulations of continuity are given. These concepts allow us to make precise when two metric spaces are essentially the same, termed homeomorphic. For example, ${]{0,1}[}$ is homeomorphic to $\mathbb {R}$ but not to $[0,1]$.