updates>!?

content/Math4202/Exam_reviews/Math4202_P1.md

@@ -14,15 +14,48 @@ A topological manifold is a topological space that satisfies the following:
 
 - (2 points) Prove that real projective space $\mathbb{R}P^2$ is a manifold.
 
-Let $\mathbb{R}P^2=\mathbb{R}^3/\sim$ where $(x,y,z)\sim(x',y',z')$ if $\lambda(x,y,z)=(x',y',z')$ for some $\lambda\in \mathbb{R}$.
+Let $\mathbb{R}P^2=\mathbb{R}^3/\sim$ where $(x,y,z)\sim(x',y',z')$ if $\lambda(x,y,z)=(x',y',z')$ for some $\lambda\in \mathbb{R},\lambda\neq 0$.
 
 1. It is Hausdorff since $\mathbb{R}^3$ is Hausdorff, subspace of Hausdorff space is Hausdorff.
 2. It has a countable basis since $\mathbb{R}^3$ has a countable basis, subspace of countable basis has countable basis.
 3. Each point of $x$ of $RP^2$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^3$. Let $p$ be an arbitrary point in $RP^2$, Consider the projection on to the tangent plane of $p$ defined as $\mathbb{R}P^2\to \mathbb{R}^2$.
 
+<details>
+<summary>Solution on class</summary>
+
+Consider $\mathbb{R} P^n$ be the lines in $\mathbb{R}^{n+1}$ through the origin.
+
+$$
+\mathbb{R}P^n=\{v\neq 0|v\in \mathbb{R}^{n+1}\}/\sim
+$$
+
+where $a\sim b$ if there exists $\lambda\in \mathbb{R},\lambda\neq 0$ such that $\lambda a=b$.
+
+$$
+S^n=\{v\in \mathbb{R}^{n+1}|||v||=1\}
+$$
+
+First we test the local euclidean structure.
+
+Consider the hemisphere cap $U_{1,+}=\{(x_1,\dots,x_{n+1})|x_1>0\}$, note that this cap induce a quotient mapping to some open set of $\mathbb{R}P^n$
+
+Note that the cap $U_{1,+}$ is local euclidean by the bijective projection map to $\mathbb{R}^n$ $(x_1,\dots,x_{n+1})\mapsto(x_2,\dots,x_{n+1})$.
+
+And with $U_{1,-},U_{2,+},U_{2,-},\dots,U_{n,+},U_{n,-}$ we can construct a open cover of $\mathbb{R}P^n$. Since for any of the point in $\mathbb{R} P^n$ we can have some non-zero coordinates that projects to $S^n$ and we can build such cap.
+
+Second we show the second countability.
+
+Take the cap with rational coordinates, and this creates a countable basis.
+
+Third we prove the Hausdorff property.
+
+Consider $x=(x_1,\dots,x_{n+1})\in \mathbb{R}P^n$, $y=(y_1,\dots,y_{n+1})\in \mathbb{R}P^n$.
+
+</details>
+
 - (2 points) Find a 2-1 covering space of $RP^2$.
 
-Take $\mathbb{R}P^2\to S^2$ by $x\to x/\|x\|$.
+Take $\mathbb{R}P^2\to S^2$ with quotient topology where $v\sim -v$.
 
 ## Problem 2