dimensiontopological

Dimensiontopological is a term used in mathematical literature to denote the study of the interplay between dimension theory and topology. It treats dimension not merely as a static attribute of a space but as a property that interacts with the topological structure, maps, and subspaces.

Overview and scope: The notion emphasizes how different definitions of dimension—such as covering dimension, small and large inductive dimension, and, in some settings, cohomological dimension—vary under continuous mappings, products, and subspaces. The aim is to provide a unified language to compare dimensional behavior across spaces and constructions.

Key concepts: A central object is a dimension function d(X) taking values in the nonnegative integers or infinity. Central results concern monotonicity under continuous images, products bounds d(X×Y) in terms of d(X) and d(Y), and dimension-raising or dimension-lowering phenomena under mappings with specific properties. The framework often uses open covers, refinements, nerves of covers, and cohomological tools to establish bounds and equalities. For separable metric spaces, relationships among covering, inductive, and cohomological dimensions are a common focus.

Examples and scope: Classical spaces illustrate the theory—Euclidean n-space has dimension n, the Cantor set is zero-dimensional, and the Hilbert cube has a fixed separable structure with known dimension. Dimensiontopological also investigates spaces where topological dimension and intuitive geometric dimension diverge, as well as stability results under products and limits.

Relation to other fields: As a cross-disciplinary area, it informs topology, geometric group theory, and analysis, and relates to data topology and manifold theory. Notable related topics include dimension theory, covering dimension, inductive dimension, and cohomological dimension.