diff --git a/content/Math4201/Math4201_L29.md b/content/Math4201/Math4201_L29.md index 4c91914..de9bd65 100644 --- a/content/Math4201/Math4201_L29.md +++ b/content/Math4201/Math4201_L29.md @@ -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 $x$, $(U-\{x\})\cap A\neq \emptyset$._ +_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 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 > **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) > > **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). > > [S000103](https://topology.pi-base.org/spaces/S000103)