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

#### Peter

##### Well-known member
MHB Site Helper
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

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter ,

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

#### Peter

##### Well-known member
MHB Site Helper
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

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter ,

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

#### Peter

##### Well-known member
MHB Site Helper
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

#### GJA

##### Well-known member
MHB Math Scholar
Yes, this is correct. Nicely done.

#### Peter

##### Well-known member
MHB Site Helper
Yes, this is correct. Nicely done.

Thanks for all your help, GJA ...

It is much appreciated...

Peter