In this article, we start by a review of the circle group
[1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle
. Using points on
under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted
of projection matrices. Together with the induced topology, it will be demonstrated that
is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on
to generate subgroups of
.