Topologically

Topologically is an adverb used to describe properties, statements, or arguments that pertain to topology rather than to concrete geometric measurements. In mathematics, topology studies notions of closeness, continuity and convergence in a broad sense, focusing on how spaces are connected and how they can be shaped without tearing or gluing.

A central concept is the topological space, which consists of a set equipped with a topology—an array of open subsets that satisfy certain axioms about unions and finite intersections. A function between topological spaces is continuous if the preimage of every open set is open. Two spaces are topologically equivalent, or homeomorphic, when there exists a bijective continuous map with a continuous inverse; such a map indicates that the spaces can be deformed into one another without cutting or attaching new parts.

Topological properties are those preserved under homeomorphism, such as connectedness, compactness, and the number of connected components. These properties depend on the global, qualitative structure of a space rather than on precise geometric measurements like distances or angles. For example, a coffee mug with one handle is topologically equivalent to a torus, since each can be continuously deformed into the other without tearing.

The term topologically is frequently used to contrast qualitative, deformation-based properties with metric properties. It appears in many areas of mathematics and science, including geometry, physics, and data analysis, where topological ideas help capture invariant features under continuous change, culminating in fields such as topological data analysis.