# Thread: Relatively Open Sets ... Stoll, Theorem 3.1.16 (a) ...

1. I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of Theorem 3.1.16

Stoll's statement of Theorem 3.1.16 and its proof reads as follows:

If $\displaystyle U = X \cap O$ for some open subset $\displaystyle O$ of $\displaystyle \mathbb{R}$ ...

... then ...

... the subset $\displaystyle U$ of $\displaystyle X$ is open in $\displaystyle X$ ...

Help will be much appreciated ...

My thoughts so far as as follows:

Suppose $\displaystyle U = X \cap O$ for some open subset $\displaystyle O$ of $\displaystyle \mathbb{R}$ ...

Need to show $\displaystyle U$ is open in $\displaystyle X$ ... that is for every $\displaystyle p \in U$ there exists $\displaystyle \epsilon \gt 0$ such that $\displaystyle N_{ \epsilon } (p) \cap X \subset U$ ... ...

Now ... let $\displaystyle p \in U$ ...

then $\displaystyle p \in O$ ...

Therefore there exists $\displaystyle \epsilon \gt 0$ such that $\displaystyle N_{ \epsilon } (p) \subset O$ ... since $\displaystyle O$ is open ...

BUT ...

... how do I proceed from here ... ?

Hope someone can help ...

Peter  Reply With Quote

2.

3. Hi Peter,

Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?  Reply With Quote Originally Posted by GJA Hi Peter,

Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?

Hi GJA ...

Still perplexed ... can you help further...

Peter  Reply With Quote

5. Hi Peter,

Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.  Reply With Quote Originally Posted by GJA Hi Peter,

Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.

Thanks GJA ...

I think the argument you're suggesting is as follows:

We have $N_{\epsilon}(p)\subset O$

So therefore $\displaystyle N_{\epsilon}(p) \cap X \subset O \cap X$ ...

... that is $\displaystyle N_{\epsilon}(p) \cap X \subset U$ ... as required ...

Is that correct?

Peter  Reply With Quote

7. Yes, this is correct. Nicely done.  Reply With Quote Originally Posted by GJA Yes, this is correct. Nicely done.

Thanks for all your help, GJA ...

It is much appreciated...

Peter  Reply With Quote

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