Topolog

Topolog is a branch of mathematics that studies properties of spaces that are preserved under continuous deformations, such as stretching and bending but not tearing or gluing. In English, the field is called topology. The term topolog may appear as a transliteration in some languages or as a misspelling in English.

This article covers core ideas: A topological space consists of a set equipped with a topology, a collection of subsets called open sets that contain the empty set and the whole set and are closed under arbitrary unions and finite intersections. A function between topological spaces is continuous if the preimage of every open set is open. A map with a continuous inverse is a homeomorphism, indicating two spaces are topologically the same.

Topologies give rise to concepts such as subspace topology, product topology, and quotient topology. Important themes include separation axioms (T0, T1, Hausdorff), connectedness vs. disconnectedness, compactness, and convergence that can be described by sequences, nets, or filters. General topology concerns the properties of spaces themselves, while algebraic and geometric topology study spaces via algebraic invariants or geometric structures.

Classic examples include the real numbers with the standard topology, the discrete topology (where every subset is open), and the indiscrete topology (only empty set and whole set are open).

Topolog has influenced many areas of mathematics and science, including analysis, geometry, and data science, where the shape of data is studied via topological methods. The field continues to develop through new invariants, techniques, and connections to other disciplines.