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Update Math4201_L29.md

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@@ -55,6 +55,8 @@ A space $X$ is limit point compact if any infinite subset of $X$ has a [limit po
_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 $x$, $(U-\{x\})\cap A\neq \emptyset$._ _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 $x$, $(U-\{x\})\cap A\neq \emptyset$._
_This property also holds for finite sets, for example, any finite set with discrete topology is limit point compact. (since you cannot find a infinite subset of a finite set that has a limit point)_
#### Definition of sequentially compact #### Definition of sequentially compact
A space $X$ is sequentially compact if any sequence has a convergent subsequence. i.e. If $\{x_n\}_{n\in\mathbb{N}}$ is a sequence in $X$, then there are $n_1<n_2<\dots<n_k<\dots$ such that $\{y_i=x_{n_i}\}_{i\in\mathbb{N}}$ is convergent. A space $X$ is sequentially compact if any sequence has a convergent subsequence. i.e. If $\{x_n\}_{n\in\mathbb{N}}$ is a sequence in $X$, then there are $n_1<n_2<\dots<n_k<\dots$ such that $\{y_i=x_{n_i}\}_{i\in\mathbb{N}}$ is convergent.
@@ -128,12 +130,14 @@ That means, sequentially compact is a stronger property than limit point compact
> >
> **There exists spaces that are sequentially compact but not compact.** > **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. > [link to spaces](https://topology.pi-base.org/spaces?q=Sequentially%20Compact%2B%7ECompact)
> >
> [S000035](https://topology.pi-base.org/spaces/S000035) > [S000035](https://topology.pi-base.org/spaces/S000035)
> >
> **There exists spaces that are compact but not sequentially compact.** > **There exists spaces that are compact but not sequentially compact.**
> >
> [link to spaces](https://topology.pi-base.org/spaces?q=Compact%2B%7ESequentially%20Compact)
>
> 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). > 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) > [S000103](https://topology.pi-base.org/spaces/S000103)