Real & Complex Projective Space (Part 2)
We can now move on to talk about Complex Projective Space, using some of the intuition we built in the real case. Define the equivalence relation on , so that if with . Then define . We can still think of as the space of complex lines through the origin in , but because multiplication by complex numbers “rotate” as well as “scale,” it’s not quite as easy to see what’s going on here.
In this post, we will prove a number of nice topological properties about , and in the last post we will attempt a simple study of its structure, introducing important examples.
1. is compact.
Let . This is the -dimensional complex sphere. Next, define the projection map . Note that for every non-zero vector in there is (at least one) scalar such that . Thus, every equivalence class intersects the complex -sphere at least once. In fact, they intersect the complex sphere in many different places, so it isn’t a transversal.
As a result, we can see that . Since is compact, and is continuous (by the construction of the quotient space), so is .
2. is open.
It will suffice to show that (the equivalence class of ) is open for any open . This is because is open iff is open (by the construction of the quotient topology), and . Write . is open, so is the union of open sets, and hence is open itself.
Before we move on, we need to prove two simple theorems about quotient spaces.
Theorem 1: Let be a topological space and let be an equivalence relation on . Let be the projection map. A function from to a topological space is continuous if and only if is continuous.
Proof: Since the composition of continuous functions is continuous, the continuity of $f$ implies the continuity of . Conversely, if is continuous, let be an open subset of . Then is an open subset of . By the definition of the quotient topology, is an open subset of . Thus is continuous.
Theorem 2: Let be a continuous function form a topological space to a topological space . Let be an equivalence relation on such that is constant on each equivalence class. Then there exists a continuous function such that .
Proof: Consider the following diagram. Define the value of on an equivalence class to be the value of at any element of the equivalence class. Then is well defined and . The continuity of follows from Theorem 1.
3.The equivalence of with .
Define on as follows: if and . Let be the natural projection map. We want to show that . By Theorem 2, we need to construct a map from to that respects .
Observe that is a bijection, as iff . By construction, is continuous. We want to show it is open, and hence a homeomorphism. We know that is open iff is open in . To see that is open, note that is open iff is open in , which is true iff is open.
Hence, it will suffice to show that if is relatively open, so is . To show this, construct a map by . This map normalizes a vector, sending it to sphere. Note that is continuous, so that if is open, is open. But , so we are done.
4. is Hausdorff.
We will work in , identifying with . Let such that . Let and . Then . Moreover, and are closed, hence compact. Let be the distance between and , and define and . These are the equivalence classes of open balls around and in , and are open themselves.
We claim that . Suppose, to the contrary, that . Then with and . Similarly, with and .
This contradicts the fact that is the infimum of the distances between points in and .
By construction, , both are open, and and , so is Hausdorff. .
In summary, we have shown that complex projective space is compact and hausdorff, and the projection map is a homeomorphism. Moreover, we can study complex projective space by working only with a quotient space of the complex sphere, which is simpler to work with.