topoloji

Topoloji, or topology in English, is the branch of mathematics that studies properties of spaces preserved under continuous deformations such as stretching and bending. It emphasizes notions of closeness, continuity, and connectivity without relying on exact measurements of distance or angles. Formally, a topological space consists of a set X together with a topology T, a collection of subsets of X called open sets, satisfying that the union of any family of open sets is open, the intersection of any finite number of open sets is open, and that both the empty set and X are open.

Key constructions include subspace topology, product topology, and quotient topology. A function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X. A homeomorphism is a bijective continuous map with a continuous inverse, defining a topological equivalence between spaces. Examples illustrating the concept range from the discrete topology, where every subset is open, to the indiscrete topology, where only the entire set and the empty set are open, and the cofinite topology, where open sets have complements that are finite.

Important properties studied in topology include convergence, compactness, and connectedness, as well as separation axioms such as T0, T1, and Hausdorff (T2) spaces. Metric spaces naturally induce topologies via open balls, illustrating how geometry and topology intersect.

Topologie has several main branches: point-set topology (foundations of open sets and continuity), algebraic topology (invariants like fundamental groups and homology), differential topology (manifolds and smooth maps), and geometric topology (manifolds and their geometric structures). It also informs applied areas such as topology-driven data analysis and various physics contexts.