updates?

content/Math4201/Math4201_L29.md

@@ -51,7 +51,9 @@ Therefore, $X$ is uncountable.
 
 #### Definition of limit point compact
 
-A space $X$ is limit point compact if any infinite subset of $X$ has a limit point.
+A space $X$ is limit point compact if any infinite subset of $X$ has a [limit point](./Math4201_L8#limit-points) in $X$.
+
+_That is, $\forall A\subseteq X$ and $A$ is infinite, there exists a point $x\in X$ such that $x\in U$, $\forall U\in \mathcal{T}$ containing $x$, $(U-\{x\})\cap A\neq \emptyset$._
 
 #### Definition of sequentially compact
 
@@ -59,7 +61,7 @@ A space $X$ is sequentially compact if any sequence has a convergent subsequence
 
 #### Theorem of limit point compact spaces
 
-If $(X,d)$ is a metric space, then the following are equivalent:
+If $(X,d)$ is a **metric space**, then the following are equivalent:
 
 1. $X$ is compact.
 2. $X$ is limit point compact.
@@ -85,6 +87,8 @@ $X$ is not sequentially compact because the sequence $\{(n,a)\}_{n\in\mathbb{N}}
 
 First, we show that 1. implies 2.
 
+We proceed by contradiction.
+
 Let $X$ be compact and $A\subseteq X$ be an infinite subset of $X$ that doesn't have any limit points.
 
 Then $X-A$ is open because any $x\in X-A$ isn't in the closure of $A$ otherwise it would be a limit point for $A$, and hence $x$ has an open neighborhood contained in the complement of $A$.
@@ -114,5 +118,22 @@ Continue with the proof that 2. implies 3. next time.
 
 Proof of 1. follows from the theorem of limit point compact spaces.
 
-
 </details>
+
+That means, sequentially compact is a stronger property than limit point compact, and compact is the stronger property than limit point compact.
+
+> [!WARNING]
+>
+> Hope you will not use it soon for your exams but here are some interesting examples.
+>
+> **There exists spaces that are sequentially compact but not compact.**
+>
+> Consider the interval $[0,1)$ with the standard topology over $\mathbb{R}$. This space is sequentially compact but not compact.
+>
+> [S000035](https://topology.pi-base.org/spaces/S000035)
+>
+> **There exists spaces that are compact but not sequentially compact.**
+>
+> Consider the space of functions $f:[0,1]\to [0,1]$ with the topology of pointwise convergence. This space is compact $I^I$ but not sequentially compact (You can always find a sequence of functions that does not converge to any function in the space, when there is uncountable many functions in the space).
+>
+> [S000103](https://topology.pi-base.org/spaces/S000103)