From 3cbe08e3fe2766f1b938450da9a6c5b1890d17ac Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 2 Mar 2026 11:53:26 -0600
Subject: [PATCH] updates>!?

---
 content/Math4202/Exam_reviews/Math4202_E1.md | 21 ++++++++---
 content/Math4202/Exam_reviews/Math4202_P1.md | 37 ++++++++++++++++++--
 2 files changed, 51 insertions(+), 7 deletions(-)

diff --git a/content/Math4202/Exam_reviews/Math4202_E1.md b/content/Math4202/Exam_reviews/Math4202_E1.md
index 3c316df..4f2801c 100644
--- a/content/Math4202/Exam_reviews/Math4202_E1.md
+++ b/content/Math4202/Exam_reviews/Math4202_E1.md
@@ -34,7 +34,7 @@ $$
 X_1=\{(e_\alpha^1,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^1\to X_0\}
 $$
 
-where $\varphi$ is a continuous map, and $e_\alpha^1$ is a $1$-cell (interval).
+where $\varphi_\alpha^1$ is a continuous map that maps the boundary of $e_\alpha^1$ to $X_0$, and $e_\alpha^1$ is a $1$-cell (interval).
 
 $$
 X_2=\{(e_\alpha^2,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^2\to X_1\}=(\sqcup_{\alpha\in A}e_\alpha^2)\sqcup X_1
@@ -74,10 +74,9 @@ $X_2=$ ballon shape with boundary of circle collapsing at $a$
 
 An $m$-dimensional **manifold** is a topological space  $X$ that is
 
-1. Hausdorff
-2. With a countable basis
-3. Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$. (local euclidean)
-
+1. Hausdorff: every two distinct points of $X$ have disjoint neighborhoods
+2. Second countable: With a countable basis
+3. Local euclidean: Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
 
 #### Whitney's Embedding Theorem
 
@@ -100,6 +99,18 @@ Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
 
 ### Homotopy
 
+#### Definition of homotopy equivalent spaces
+
+Let $f:X\to Y$ and $g:X\to Y$ be tow continuous maps from a topological space $X$ to a topological space $Y$.
+
+$f\circ g:Y\to Y$ should be homotopy to $Id_Y$ and $g\circ f:X\to X$ should be homotopy to $Id_X$.
+
+#### Definition of homotopy
+
+Let $f:X\to Y$ and $g:X\to Y$ be tow continuous maps from a topological space $X$ to a topological space $Y$.
+
+If there exists a continuous map $F:X\times [0,1]\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x\in X$, then $f$ and $g$ are homotopy equivalent.
+
 #### Definition of null homology
 
 If $f:X\to Y$ is homotopy to a constant map. $f$ is called null homotopy.
diff --git a/content/Math4202/Exam_reviews/Math4202_P1.md b/content/Math4202/Exam_reviews/Math4202_P1.md
index 2bdabdc..396b9f4 100644
--- a/content/Math4202/Exam_reviews/Math4202_P1.md
+++ b/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