Dimension (mathematics)

Dimension in mathematics is a generalization of the idea that a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. Loosely, dimension describes the number of independent freely varying parameters.

In linear algebra

Dimension of a vector space: The dimension of a vector space over a field equals the cardinality of a basis for that vector space (we can always find a basis for any vector space, using the axiom of choice, and any two bases have equal cardinality).

A field has dimension one as a vector space over itself. Dimension of a direct sum of vector spaces is the sum of their dimensions, and dimension of a tensor product of vector spaces is the product of their dimensions.

No subject wiki entry

In commutative and noncommutative algebra

Krull dimension: The Krull dimension of a commutative unital ring is the maximum possible length of an ascending chain of prime ideals in the ring.

Primary subject wiki entry: Commalg:Krull dimension

Also located at: Mathworld:KrullDimension, Planetmath:KrullDimension, Wikipedia:Krull dimension

Homological dimension of a module

Cohomological dimension of a module

Global dimension of a ring

In point-set topology

Topological dimension: The topological dimension or covering dimension or Lebesgue covering dimension of a topological space is defined as the smallest integer $m$ such that any open cover of the topological space has an open refinement that has order at most $m+1$.

Primary subject wiki entry: Topospaces:Topological dimension

Also located at: Wikipedia:Lebesgue covering dimension

Large inductive dimension

Small inductive dimension

In differential geometry

Dimension of a manifold