topologias

Topologies, in mathematics, are a way to formalize the notion of continuity and closeness without using distances. A topology on a set X is a collection T of subsets of X, called open sets, satisfying three axioms: the empty set and X belong to T; arbitrary unions of members of T belong to T; and finite intersections of members of T belong to T. The pair (X, T) is a topological space, and the elements of T are the open sets.

Examples include the indiscrete (trivial) topology {∅, X}, the discrete topology which consists of all subsets of X, the standard (Euclidean) topology on R generated by open intervals, the cofinite topology where complements of finite sets are open, and the Zariski topology in algebraic geometry.

Maps between topological spaces are continuous when the preimage of every open set is open. Homeomorphisms are continuous bijections with continuous inverses and are the isomorphisms in the category of topological spaces. Given a subset A of X, the subspace topology on A consists of intersections U∩A with U open in X. The product topology on a product X×Y is generated by products of open sets; the quotient topology arises from surjective maps.

Key concepts associated with topologies include convergence (via nets or sequences in suitable spaces), compactness, connectedness, and separability. Some topologies arise from metrics (metric topology) or from order structures (order topology). Topology has a central place in modern mathematics, with a formal category of spaces and continuous maps, often denoted Top.