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The dimension of a vector space is equal to the maximum length (number of inclusions) contained a chain of nested subspaces. The view of the dimension as chain length allows for a generalization to other structures.

For example, the Krull dimension of a commutative ring as the maximum length of a chain of prime ideals contained in each other is defined.

Likewise, the dimension of a manifold, the maximum length of a chain of interlocking contained manifolds, where each link in the chain edge of a subset of the former. For example, the edge of the earth, the earth's surface, the edge of their subset of Germany is the state border, the edge of a particular section of the border, the two end points - there is no longer chain, the earth dimension 3 Because inclusion and formation are always defined, this provides a concept of dimension for each topological space (dimension). Another topological concept of dimension is the coverage dimension.
I mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it.
Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it for example, to locate a point on the surface of a sphere you need both its latitude and its longitude. The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces. In physical terms, dimension refers to the constituent of all space and its position in time, as well as the spatial constitution of objects within —structures that have correlations with both particle and field conceptions, interact according to relative properties of mass, and which are fundamentally mathematical in description. These or other axes may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and events. Physical theories that incorporate time, such as general relativity, are said to work in 4-dimensional space time. Modern theories tend to be "higher-dimensional" including quantum field and string theories. The state-space of quantum mechanics is an infinite-dimensional function space.
In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate, so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.