Orthogonal Projection is a very familiar topic in Linear Algebra  . With reference to  , it is already known that if V is a finite-dimensional vector space and P is a projection on , where W is a subspace of V. Then P is idempotent, that is . P is the identity operator on W, that is . We also know that W is the range of P and if U is the kernel of P then . It is easy to show that . It is also a known fact that these operators are bounded i.e. . In this paper we will focus on projections in and define a different construct for these operators. Starting in the next section with the circle group  it is possible to endow the set of projective operators with a group and topological structure.
2. Notation Used in This Article
1) The unit circle as a Topological Group.
2) The circle group defined as .
3) is an element in the topology of .
4) the usual topology on .
5) the topology on the unit circle .
6) is the Quotient Metric on .
7) opens balls in .
8) is open in .
9) the circle group as a topological group.
10) the unit circle as topological group.
11) open set in generated by counter-clockwise rotation.
12) open set in generated by clockwise rotation.
13) A projection matrix at angle .
14) the topological projection manifold.
15) open sets in .
3. A Brief Review of the Circle Group 𝕋 and Its Topology
3.1. The Group Structure
The topological group as in  is constructed by viewing the real line as topological group where we identify . The topology is the usual topology induced by the metric . We then define the following equivalence relation
Without proof, we see that this is an equivalence relation. The circle group is then defined as
It is very clear that this forms a group under addition such that
With the identity
The additive inverse is simply given by such that
It is clear that this is associative and hence is a group.
3.2. The Topological Structure on 𝕋  
First, we see that we have the projection mapping such that
Since is a topological space we can define the topology on be declaring an open set to be open if and only if is open in .
Suppose we have some open then we have hence we define to be open in . It is clear that is open in . Next, we show the closure w.r.t unions and intersections. From topology, it is well known that the union of open sets is open and the intersection of open sets is also open, therefore, we can see that
For the intersections we get
Clearly, is open in since .
3.3. The Quotient Metric on the Circle Group
The quotient topology is induced by the quotient metric defined as
We can define an open ball from this metric in the following way
Let and be the kth representative of , that is for some . Let be some other point in . Then we think of . Hence, the open sets in can be defined by using the definition of the open balls (above) and the canonical projection mapping .
In order for the canonical map to make sense and in order to satisfy we construct the open balls as discussed above
This defines a topology on . Furthermore, even though is open and bounded in , any open set in can be written as the countable union of open balls .
Also we note that this metric is a pseudo-metric. A pseudo-metric is metric is similar to usual metric spaces with the exception that it possible to have the following result
In fact, this implies that
Hence, is a topological group, denoted .
4. The Mapping
We now consider the unit circle as the set
Clearly, can e endowed with the subspace topology generated by the metric such that we define the metric to be i.e. the shortest arc length between the points .
Clearly, this defines a topology on . Equipped with this topology we can say that is a topological group. The mapping
This, clearly defines an isomorphism. The group operation on the circle is given by multiplication as follows
It is clear that open balls on are mapped into open arcs on .
5. Defining Projections on
We now focus on the topology generated by open arcs. We can write
We define the following mapping
Given that each open set in the topology satisfies the metric implies that is bijective and hence has an inverse.
Moreover, we have
The group operation of the projector is defined as follows
where and respectively.
Clearly, it can be easily verified that this defines an abelian group as per definition from   . Also, this is consistent with the group operation on and shows that is a group homomorphism.
Image and Kernel
Let . The vector can be written as . denotes the angle between vector u and the x-axis. Then we have
This is just the familiar projection formula. Hence, the image is just the 1-dimensional subspace spanned by .
The Kernel, substitution of by gives us the following result
Hence, the kernel is the orthogonal complement of the subspace spanned by . It is also easy to verify that is idempotent.
6. Projections as a Topological Manifold
Clearly, the topology induces a topological structure on the projector group .
The topology is Hausdorff and Second Countable.
Suppose we have 2 points and , there exists such that and an open arc such that such that . Hence, by the group homomorphism we
Therefore, it is Hausdorff.
For the second countability property, we proceed as follows.
Starting by using a countable basis in of the form where . Since is countable then the set
Is a countable basis on . This implies that the mapping
induces a second countable basis on , which in turn, implies that induces a second countable on .
Hence, we have a topological manifold.
The topological manifold is homeomorphic to .
We can define an atlas as follows
The mapping is bijective since
Since we can get the following result
We get the following matrix
Now it is easy to show that
Therefore, we have
However, since is bounded such that implies that is bijective on each . It is easy to see that is also continuous hence it defines a homeomorphism. For the transition functions, we arbitrarily choose some such that then we have
Is open in .
Theorem 3. The topological manifold is a Lie Group.
Since the group operation is addition of angles and implies that is both continuous and . Same argument applies to the inverse mapping. It also clear that elements of these matrices are smooth transcendental functions of which are also . Hence, we have a Lie Group. ■
7. Group Action of ℤ on
We now define the following group action of on the Lie Group in the following way
Lemma 4. Let be some point in the circle group for some fixed . Then , is a subgroup of .
The action of on given by generates a subgroup since
With the additive identity and additive inverse in , we have a subgroup of . ■
In conclusion, we have demonstrated the link between the circle group, the circle and the projection group. There is much to do to continue developing the theory. We wish to continue on this topic in subsequent articles.
I wish to extend my gratitude for the all, the very helpful advice from Prof. Nigel Atkins from Kingston University who was kind enough to take my numerous phone calls despite his busy teaching schedule.