Topology is a branch of mathematics that involves properties that are preserved by continuous transformations. In fact, a “topology” is precisely the minimum structure on a set that allows one to even define what “continuous” means. Continuity, which refers to changes that may “stretch” or “fold” but never “tear”, is a fundamental concept in mathematics and science. By establishing the “minimal” requirements to understand continuity, topology has applications in almost every branch of mathematics and science.
Topology itself has many active branches, many of which are intimately connected to other areas of mathematics and science: knot theory (biology, physics, algebra), algebraic topology (differential geometry, algebra, data analysis, physics, chemistry, engineering, robotics), geometric group theory (algebra).