updates>!?

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.