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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Stoll - Theorem 3.1.16 ... .png


Can someone please help me to demonstrate a formal and rigorous proof of the following:


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
Jan 16, 2013
261
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
Jun 22, 2012
2,891
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
Jan 16, 2013
261
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
Jun 22, 2012
2,891
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
Jan 16, 2013
261
Yes, this is correct. Nicely done.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891