Topological Compactness and Compactness of a Set

**meks08999** · September 21st 2009, 05:43 PM

I need help in each of the problems. See attached picture.

Thanks

**redsoxfan325** · September 21st 2009, 09:43 PM

Originally Posted by meks08999

I need help in each of the problems. See attached picture.

Thanks

9. Let $\{U_i\}_{i\in I}$ be an open cover of $F$ and let $U_k\in\{U_i\}_{i\in I}$ be the set that contains $x_0$ . Since $x_0$ is a cluster point, $U_k$ also contains all but finitely many $x_n$ . Assume that it covers all but $N$ points in $\{x_n\}$ .Thus there are at most $N$ more open sets in $\{U_i\}_{i\in I}$ (in addition to $U_k$ ) that completely cover $\{x_n\}$ , and so you have proved that every open cover has a finite subcover, and $F$ is compact.

I'm not sure about 10. I know in a compact space every sequence has a convergent subsequence, but I don't know how to find two of them.

**Enrique2** · September 22nd 2009, 12:16 AM

For 10 you have to show that there are at least two cluster points in the sequence. Assume that there is $\alpha$ only one cluster point

1. First, show that there is $\varepsilon$ such that there are infiniely many $(x_n)$ 's outside of $B(\alpha,\varepsilon)$ . If such an $\varepsilon$ does not exist we would have that the sequence converge to $\alpha$

2. For the subsequence formed by the $(x_n)$ 's outside of $B(\alpha,\varepsilon)$ , observe that it is in the compact set $K$ . Hence there exists a cluster point which can't be $\alpha$ .

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