partial updates for review, and fix typos

content/Math4201/Math4201_L15.md

@@ -72,9 +72,13 @@ Find an example of a function $f:X\to Y$ which is not continuous but for any con
 <details>
 <summary>Solution</summary>
 
-Let $f:S^1\to [0,1)$ be defined by $f(x,y)=\sin^{-1}(\frac{y}{x})$.
+Consider $X=\mathbb{R}$ with complement finite topology and $Y=\mathbb{R}$ with the standard topology.
 
-This is not continuous because $[0,1)$
+Take identity function $f(x)=x$.
+
+This function is not continuous by trivially taking $(0,1)\subseteq \mathbb{R}$ and the complement of $(0,1)$ is not a finite set, so the function is not continuous.
+
+However, for every convergent sequence in $X$, $\{x_n\}_{n=1}^\infty\to x$, the sequence $\{f(x_n)\}_{n=1}^\infty\to f(x)$ trivially.
 
 </details>