topologies

In mathematics, a topology on a set X is a collection T of subsets of X, called open sets, that includes the empty set and X, and is closed under arbitrary unions and finite intersections. The pair (X, T) is called a topological space. The complements of open sets are closed sets.

Continuity: a function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.

Subspace topology: for A ⊆ X, T_A = {U ∩ A: U ∈ T}. Product topology: on ∏i Xi, it is the coarsest topology making the projections continuous; a basis consists of sets ∏i Ui with Ui open in Xi and Ui = Xi for all but finitely many i. Quotient topology: q: X → Y gives the topology on Y where V is open iff q^{-1}(V) is open in X.

Bases and subbases: a collection B is a basis if every open set is a union of elements of B; a collection S is a subbasis, generating a topology by finite intersections and unions. Examples include discrete, indiscrete, the standard topology on R, and the cofinite topology.

Common properties and notions: separation axioms T0, T1, T2; connectedness, compactness; regularity and normality; many concepts are defined via open sets, closures, and interiors. If X is a metric space with d, the metric topology induces its standard topology.

Category and convergence: topological spaces and continuous maps form a category called Top; convergence can be described with nets or filters, generalizing sequences. Topology provides the language for continuity, convergence, and compactness in analysis and geometry.