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In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.

Definition (topology)

A topological space X is said to be equidimensional if for all points p in X, the dimension at p, that is dim p(X), is constant. The Euclidean space is an example of an equidimensional space. The disjoint union of two spaces X and Y (as topological spaces) of different dimension is an example of a non-equidimensional space.

Definition (algebraic geometry)

A scheme S is said to be equidimensional if every irreducible component has the same Krull dimension. For example, the affine scheme Spec k/(xy,xz), which intuitively looks like a line intersecting a plane, is not equidimensional.

Cohen–Macaulay ring

An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.

References

1. Wirthmüller, Klaus. A Topology Primer: Lecture Notes 2001/2002 (PDF). p. 90. Archived (PDF) from the original on 29 June 2020.
2. Sawant, Anand P. Hartshorne's Connectedness Theorem (PDF). p. 3. Archived from the original (PDF) on 24 June 2015.

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

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