Topology

Topology keeps continuity and throws away distance, or rather, it notices that continuity can be stated without numbers. You declare which sets are open. The declaration must include the empty set and the whole space, and it must survive arbitrary unions and finite intersections. In the real line the open sets are unions of open intervals, and that choice reproduces the ε–δ notion of continuity. Other choices of open sets give other geometries: the discrete topology makes every function continuous, the indiscrete topology makes only constants continuous into a Hausdorff space.

With open sets in hand, continuity is a single sentence: f is continuous when f⁻¹(U) is open whenever U is open. Sequences are no longer the definition, though in metric spaces they still detect the same closed sets. A homeomorphism is a continuous bijection with continuous inverse. It is the notion of “the same shape” for a topologist. A coffee cup with a handle and a doughnut are homeomorphic; a coffee cup and a ball are not, because removing a circle can disconnect one and not the other in a way no homeomorphism can repair. That invariant is the start of algebraic topology.

Two properties do most of the work that closed intervals did in analysis. A space is compact when every cover by open sets has a finite subcover. A continuous image of a compact space is compact, so a continuous real function on a compact set attains its bounds — the extreme value theorem, with the interval hypothesis replaced by the property that made the proof work. A space is connected when it is not the union of two nonempty disjoint open sets. Continuous images of connected sets are connected, which is the intermediate value theorem in its natural habitat.