In any case the dimension of a set is the number of numbers needed to define any element of the set.

Consider the set of points

\[\{ (x,x^2) : x \in \mathbb{R} \}\]

. This set defines the line \[y=x^2\]

which is a curve in the plane. The plane is two dimensional but the curve is one dimensional since only one point \[x\]

is needed to define each point on the curve.This example shows how it is possible for a space to be embedded in a higher dimensional subspace.

The example is illustrative.

\[x\]

and \[y\]

are coordinates on different axes. The coordinates are not independent if one coordinate is a function of the other, or if some coordinates are functions are some other coordinates. Some physical systems are defined by physics properties.One such is a confined ideal gas. A gas is defined totally by its internal heat energy

\[U\]

, its temperature \[T\]

, its pressure \[p\]

and volume \[V\]

.These quantities are not independent however.

\[U\]

is directly related to \[T\]

by the equation \[U= \frac{3}{2} kT\]

and we can define \[T\]

and hence \[U\]

in terms of \[p\]

and \[V\]

by the equation \[pV=nRT\]

where

\[n\]

is the number of mols and \[R=8.314 J/mol/K\]

is the Universal molar gas constant.\[U\]

and \[T\]

are not needed to describe the state of a gas. We only need \[p\]

and \[V\]

- or in fact any two of \[U, \: T, |; p, V\]

. The dimension of the state space of an ideal gas is 2 and we can plot any state of an ideal gas in the plane.