Topology is the study of properties of objects that do not depend on geometric measurements, and that do not change when the object is stretched or distorted without tearing. Topology is divided into three subdisciplines, which are largely different from each other: point-set topology, algebraic topology, and differential topology.

Point-set topology is the broadest and most fundamental of the different types of topology, and also the most axiomatic. The basic notion in point-set topology is the idea of continuity. The most classic definition of continuity is the one that arises in single-variable calculus, concerning functions that map the real numbers to themselves: a function is continuous, loosely speaking, if its graph has no jumps. This definition can be extended to a more general setting, for any function that maps one topological space to another. These more general functions are called continuous when they map one space to the...