For the sake of completeness, let us start with the definition of an abstract manifold.

Let \(X\) be a topological space. A **chart** (or coordinate chart) is an open set \(U\subseteq X\)
together with a homeomorphism \(\phi: U \to \mathbb{R}^n\). An **atlas** is a collection of charts \( {(U_{\alpha},\phi_{\alpha})} \) with the condition
\( X = \cup_{\alpha\in A} U_{\alpha} \). A **smooth atlas** is an atlas such that whenever \( U_{\alpha} \cap U_{\beta} \neq \emptyset \), the composition
\( \phi_{\beta} \circ \phi_{\alpha}^{-1} : \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \phi_{\beta}(U_{\alpha}\cap U_{\beta}) \) i.e. every *transition map* is smooth.
Two smooth atlases are **equivalent** if their union is also a smooth atlas. This gives a natural equivalence class of smoothly equivalent atlases
which we call a **smooth structure**. More specifically, define an equivalence relation \( \sim \) where \( P \sim Q \) if \(P, Q\) are smooth atlases that are
equivalent. The equivalence class generated by this relation is called a smooth structure.

A **topological manifold** \(X\) is a Hausdorff, second countable topological space \(X\) for which one can find an atlas. A **smooth manifold**
\(X\) is a topological manifold with a distinguished smooth structure.

Before we talk about the real projective space \( \mathbb{R}P^n \), we talk about the \(n = 2\) case.

### Definition (Real Projective Plane)

Let \(\mathbb{R}P^2\) denote the set of lines in \(\mathbb{R}^3\) that go through the origin. We say that
\(\mathbb{R}P^2\) is the **real projective plane**.

To show this set is an abstract manifold, we first have to define a topology on it, which we do using the quotient topology. To do this consider \(\mathbb{R}^3\) and the equivalence relation \(\sim\) on it where \(x\sim y\) if and only if \(x = \lambda y\) for \(\lambda\in\mathbb{R},\lambda\neq 0.\) Now, consider the mapping \(p(x\in \mathbb{R}^3-{0}) = [x]\). Call \(Y\) the set mapped to by \(p\) and say that a set \(V\) in \(Y\) is open if and only if \(p^{-1}(V)\) is open in \(\mathbb{R}^3-0\). Then, \(p\) is clearly a quotient map so that \(Y\) is given the quotient topology. We can clearly identify \(\mathbb{R}P^2\) with \(Y\) by the mapping \(f: Y \to \mathbb{R}P^2\) where \(f(x) = L(x)\) where \(L(x)\) is the line that goes through the points \(x\) and \(0\). Via, this identification, we can say a set \(V\) in \(\mathbb{R}P^2\) is open if and only if \(f^{-1}(V)\) is open. Since \(Y\) and \(\mathbb{R}P^2\) are equivalent, we could have also defined \(\mathbb{R}P^2\) to simply be the topology given by quotient \((\mathbb{R}-0)/\sim\).

For each line, identify two antipodal points each magnitude 1 from the origin. This defines the set \(S^2\). A simple figure is provided below.

### How does this fit in with my previous understanding of coordinates?

You may have thought coordinates as something which was *implicitly* connected to the space
you were working in (e.g. \(\mathbb{R}, \mathbb{R}^2, …\)) in the way that giving a
coordinate was the same as pointing out a unique place on the space. This is in fact, still
correct. In this case, let \(X = \mathbb{R}^n\) and if our neighbourhood \(U \subset X\)
is of dimension \(k\), then we let it be the identity mapping.