Merge branch 'main' of https://git.trance-0.com/Trance-0/NoteNextra-origin

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -191,6 +191,12 @@ $$
 
 ### Covering space
 
+#### Definition of partition into slice
+
+Let $p:E\to B$ be a continuous surjective map. The open set $U\subseteq B$ is said to be evenly covered by $p$ if it's inverse image $p^{-1}(U)$ can be written as the union of **disjoint open sets** $V_\alpha$ in $E$. Such that for each $\alpha$, the restriction of $p$ to $V_\alpha$ is a homeomorphism of $V_\alpha$ onto $U$.
+
+The collection of $\{V_\alpha\}$ is called a **partition** $p^{-1}(U)$ into slice.
+
 #### Definition of covering space
 
 Let $p:E\to B$ be a continuous surjective map.
@@ -225,3 +231,7 @@ Recall from previous lecture, we have unique lift for covering map.
 Let $p: E\to B$ be a covering map, and $e_0\in E$ and $p(e_0)=b_0$. Any path $f:I\to B$ beginning at $b_0$, has a unique lifting to a path starting at $e_0$.
 
 Back to the circle example, it means that there exists a unique correspondence between a loop starting at $(1,0)$ in $S^1$ and a path in $\mathbb{R}$ starting at $0$, ending in $\mathbb{Z}$.
+
+#### Theorem for induced homotopy for fundamental groups
+
+Suppose $f,g$ are two paths in $B$, and suppose $f$ and $g$ are path homotopy ($f(0)=g(0)=b_0$, and $f(1)=g(1)=b_1$, $b_0,b_1\in B$), then $\hat{f}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ and $\hat{g}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ are path homotopic.

content/Math4202/Math4202_L10.md

@@ -88,7 +88,7 @@ $\bar{f}_t=\bar{f}(1-ts)$ $s\in[\frac{1}{2},1]$.
 
 > [!CAUTION]
 >
-> Homeomorphism does not implies homotopy automatically.
+> Homeomorphism does not implies homotopy automatically. Homeomorphism doesn’t force a homotopy between that map and the identity (or between two given homeomorphisms).
 
 #### Definition for the fundamental group
 

content/Math4202/Math4202_L4.md

@@ -17,6 +17,7 @@ An $m$-dimensional **manifold** is a topological space  $X$ that is
 > Try to find some example that satisfies some of the properties above but not a manifold.
 
 1. Non-Hausdorff
+   - Real line with two origin, as discussed in homework problem
 2. Non-countable basis
    - Consider $\mathbb{R}^\delta$ where the set is $\mathbb{R}$ with discrete topology. The basis must include all singleton sets in $\mathbb{R}$ therefore $\mathbb{R}^\delta$ is not second countable.
 3. Non-local euclidean