topologytopological

Topologytopological is not a standard term in mathematics. The article below outlines the core ideas of topology and the related adjective topological, which together cover the intended concept.

Topology 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. The central object is a topological space, defined as a set X equipped with a topology τ, a collection of subsets of X called open sets that includes the empty set and X, and is closed under arbitrary unions and finite intersections. A subset U of X is open if it belongs to τ. A function f: X → Y between topological spaces is continuous if the preimage of every open set is open. Two spaces are topologically equivalent if there is a bijection between them that is a homeomorphism; that is, a function that's continuous with continuous inverse. Such homeomorphisms identify topological properties, or invariants, that do not change under deformation.

Common constructions include the subspace topology (open sets are intersections with open sets of the ambient space), the product topology (on Cartesian products), and the quotient topology (via surjective maps). Classic examples: the real numbers with the standard Euclidean topology, the discrete topology (all subsets open), and the indiscrete topology (only the empty set and the whole space).

Topological properties include connectedness, compactness, convergence, and the general notion of continuity. The adjective topological describes properties invariant under homeomorphism. The field contains subdisciplines such as general topology, algebraic topology, differential topology, and geometric topology, with applications in physics, computer science, and data analysis. In standard usage, topology and the adjective topological describe the subject and its properties; there is no widely used concept named topologytopological.