topdimensional

Topdimensional is an adjective used in several areas of mathematics to describe objects that have the maximum possible dimension within a given ambient space or within a stratified decomposition. An object is topdimensional if its dimension equals the dimension of the surrounding space or the largest dimension among the components in the decomposition.

In differential geometry and topology, a submanifold or subset of a d-dimensional ambient manifold is called topdimensional when its dimension is d. For stratified spaces, topdimensional strata are the components whose dimension equals the space’s maximal dimension; these strata typically form a dense open subset under suitable conditions and carry the most geometric information about the space.

In polyhedral and CW-geometry, topdimensional cells or faces have dimension equal to the dimension of the ambient polyhedron or complex. For example, in a d-dimensional polytope, the interior is composed of topdimensional d-dimensional cells.

In algebraic geometry, the phrase is used with varieties: an irreducible component is topdimensional if its dimension equals the dimension of the entire variety. Topdimensional components contribute to invariants like the degree and to constructs such as top-dimensional cycles in intersection theory.

In geometric measure theory and analysis, a set may be described as having topdimensional Hausdorff dimension if its dimension matches the ambient space, indicating full-dimensional geometric behavior.

See also: dimension, stratification, top-dimensional homology, top-dimensional cells.