Equidimensionality
Encyclopedia
In mathematics
, especially in topology
, equidimensionality is a property of a space that the local dimension is the same everywhere.
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 cedes an example of a non-equidimensional space.
An algebraic variety
whose coordinate ring is a Cohen–Macaulay ring is equidimensional.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, equidimensionality is a property of a space that the local dimension is the same everywhere.
A topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is said to be equidimensional if for all points p in X the dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
at p that is, dim p(X) is constant. The Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is an example of an equidimensional space. The disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of two spaces X and Y (as topological spaces) of different dimension cedes an example of a non-equidimensional space.
An algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
whose coordinate ring is a Cohen–Macaulay ring is equidimensional.