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#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

Hello everyone!

I was reading Rudin's proof for the theorem that states that the closure of a set is closed. I'll write the proof and the parts I'm having trouble connecting:

if $p\in X$ and $p\notin E$ then $p$ is neither a point of $E$ nor a limit point of $E$. (So far so good).

Hence, $p$ has a neighborhood which does not intersect $E$. (Great)

...

The compliment of $\overline{E}$ is therefore open. Hence $\overline{E}$ is closed.

Now, this is what I believe should go into the dots, but still not enough to conclude the proof:

The neighborhood around $p$ does not intersect $E$ so it lies completely in $E^c$.

Now, I can't complete the proof.

Any help is appreciated!

I was reading Rudin's proof for the theorem that states that the closure of a set is closed. I'll write the proof and the parts I'm having trouble connecting:

if $p\in X$ and $p\notin E$ then $p$ is neither a point of $E$ nor a limit point of $E$. (So far so good).

Hence, $p$ has a neighborhood which does not intersect $E$. (Great)

...

The compliment of $\overline{E}$ is therefore open. Hence $\overline{E}$ is closed.

Now, this is what I believe should go into the dots, but still not enough to conclude the proof:

The neighborhood around $p$ does not intersect $E$ so it lies completely in $E^c$.

Now, I can't complete the proof.

Any help is appreciated!

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