Identifications

A convenient way to represent ${\mathbb{S}}^1$ is obtained by identification, which is a general method of declaring that some points of a space are identical, even though they originally were distinct.^4.6 For a topological space $X$, let $X{/\sim}$ denote that $X$ has been redefined through some form of identification. The open sets of $X$ become redefined. Using identification, ${\mathbb{S}}^1$ can be defined as $[0,1]{/\sim}$, in which the identification declares that 0 and $1$ are equivalent, denoted as $0 \sim 1$. This has the effect of ``gluing'' the ends of the interval together, forming a closed loop. To see the homeomorphism that makes this possible, use polar coordinates to obtain $\theta \mapsto (\cos 2 \pi \theta,\sin 2 \pi \theta)$. You should already be familiar with 0 and $2 \pi$ leading to the same point in polar coordinates; here they are just normalized to 0 and $1$. Letting $\theta$ run from 0 up to $1$, and then ``wrapping around'' to 0 is a convenient way to represent ${\mathbb{S}}^1$ because it does not need to be curved as in (4.5).

It might appear that identifications are cheating because the definition of a manifold requires it to be a subset of ${\mathbb{R}}^m$. This is not a problem because Whitney's theorem, as mentioned previously, states that any $n$-dimensional manifold can be embedded in ${\mathbb{R}}^{2n+1}$. The identifications just reduce the number of dimensions needed for visualization. They are also convenient in the implementation of motion planning algorithms.