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#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

Hello everyone!

I'm trying to find a set a closed set ${A_n}$ whose inifinte union is not closed. Now, I can picture the following:

If I let $A_n=[-\frac{1}{n}, \frac{1}{n}]$, then $A_n$ is closed, but their union is not, simply because the point $x=0$ seems to be a limit point at """""infinity"""", but it is not in any of the $\{A_n\}$, so the union is therefore not closed.

Now, of course, I'm having trouble showing that, because, I can't always find an $r>0$ s.t. the neighborhood of center $0$ and radius $r$ intersects any of the $\{A_n\}$s.

Similarly, I'm having trouble proving that, if $B_n = [-1+\frac{1}{n}, 1-\frac{1}{n}]$, then their infinite union is not closed.

I would appreciate any help in this...

I'm trying to find a set a closed set ${A_n}$ whose inifinte union is not closed. Now, I can picture the following:

If I let $A_n=[-\frac{1}{n}, \frac{1}{n}]$, then $A_n$ is closed, but their union is not, simply because the point $x=0$ seems to be a limit point at """""infinity"""", but it is not in any of the $\{A_n\}$, so the union is therefore not closed.

Now, of course, I'm having trouble showing that, because, I can't always find an $r>0$ s.t. the neighborhood of center $0$ and radius $r$ intersects any of the $\{A_n\}$s.

Similarly, I'm having trouble proving that, if $B_n = [-1+\frac{1}{n}, 1-\frac{1}{n}]$, then their infinite union is not closed.

I would appreciate any help in this...

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