Find an infinite collection {Sn : n is an element of N} of compact sets R such that the union from n=1 to infinity Sn is compact.

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- Mar 2nd 2009, 03:13 PMnoles2188Compact sets
Find an infinite collection {Sn : n is an element of N} of compact sets R such that the union from n=1 to infinity Sn is compact.

- Mar 2nd 2009, 03:41 PMPlato
Are you sure that you have copied this correctly? As written, it is almost trivial.

$\displaystyle S_n = \left[ {\frac{{ - 1}}{n},\frac{1}{n}} \right],\;\;N = \mathbb{Z}^ + $ - Mar 2nd 2009, 04:02 PMnoles2188
I had the proper equation all written out but it didn't transfer over to the post, so I wrote it out in words. I'm sure that's how it reads in words.

- Mar 2nd 2009, 04:33 PMPlato
- Mar 2nd 2009, 04:47 PMnoles2188
In fact, I just read it again and the end should read "not compact" instead of "compact". Sorry for the confusion.

- Mar 2nd 2009, 09:02 PMaliceinwonderland
Each singleton set is compact in the standard topology on R.

Take an infinite union of singleton sets, let's say, $\displaystyle \bigcup_{k \in N}\{k\}$ is an infinite union of compact sets, which is a set of natural numbers N.

Take an open interval, let's say, $\displaystyle I_x=(x-1/2, x+1/2), x \in N$. Then $\displaystyle C=\{I_x:x \in N\}$ is an open cover for N, but it has no finite subcover.

You can find other examples in other topological spaces, such as a discrete topology on N.