Topological space

A set equipped with a collection of subsets called open sets, containing the empty set and whole set and closed under arbitrary unions and finite intersections. These axioms formalize continuity and neighborhood without requiring distance.

Open set: A member of a topology, used to describe which regions count as open. Open sets are the defining data from which a topological space is built.
Continuous function: A function between topological spaces whose inverse image of every open set is open. This definition makes continuity depend only on the open-set structures of domain and codomain.
Hausdorff space: A topological space in which any two distinct points have disjoint open neighborhoods. The Hausdorff condition strengthens how the topology separates distinct points.
Euclidean space: A real coordinate space equipped with its usual topology, generated by open balls. Euclidean space is a central example whose topology comes from ordinary distance.
Compact space: A topological space in which every open cover has a finite subcover. Compactness turns arbitrary open covers into finite data and supports powerful existence results.
Basis (topology): A collection of open sets from which every open set in a topology can be formed by unions. A basis can specify the same topology with a smaller collection of sets.
Homeomorphism: A bijective continuous function whose inverse is also continuous. Homeomorphisms identify spaces with the same topological structure.
Discrete space: A topological space in which every subset is open. It is the finest topology on a given set, unlike structures with fewer open sets.
Circle (mathematics): The set of points at a fixed distance from a center in the plane, viewed with its usual topology. The circle exemplifies a space whose global shape differs from a line despite local resemblance.
Connected space: A topological space that cannot be expressed as the union of two disjoint nonempty open subsets. Connectedness detects whether the topology splits a space into separate pieces.

Linked from 36 pages