updates

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -78,6 +78,27 @@ An $m$-dimensional **manifold** is a topological space  $X$ that is
 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 $\mathbb{R}^m$.
 
+<details>
+<summary>Example of space that is not a manifold but satisfies part of the definition</summary>
+
+Non-hausdorff:
+
+Consider the set with two origin $\mathbb{R}\setminus\{0\}$. with $\{p,q\}$, and the topology defined over all the open intervals that don't contain the origin, with set of the form $(-a,0)\cup \{p\}\cup (0,a)$ for $a\in \mathbb{R}$ and $(-a,0)\cup \{q\}\cup (0,a)$.
+
+---
+
+Non-second-countable:
+
+Consider the long line $\mathbb{R}\times [0,1)$
+
+---
+
+Non-local-euclidean:
+
+Any 1-dimensional CW complex (graph) that has a vertex with 3 or more edges connected to it will be Hausdorff and second-countable, but not locally Euclidean at those vertices.
+
+</details>
+
 #### Whitney's Embedding Theorem
 
 If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$.
@@ -97,6 +118,12 @@ Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair o
 
 Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
 
+#### Definition of paracompact space
+
+Locally finite: $\forall x\in X$, $\exists$ open $x\in U$ such that $U$ only intersects finitely many open sets in $\mathcal{B}$.
+
+A space $X$ is paracompact if every open cover $A$ of $X$ has a **locally finite** refinement $\mathcal{B}$ of $A$ that covers $X$.
+
 ### Homotopy
 
 #### Definition of homotopy equivalent spaces
@@ -128,7 +155,6 @@ Two pathes $f$ and $f'$ are path homotopic if
 
 The $\simeq$, $\simeq_p$ are both equivalence relations.
 
-
 #### Definition for product of paths
 
 Given $f$ a path in $X$ from $x_0$ to $x_1$ and $g$ a path in $X$ from $x_1$ to $x_2$.