
MHB Master
#1
January 10th, 2020,
02:09
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:
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

January 10th, 2020 02:09
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#2
January 10th, 2020,
21:01
Hi Peter,
Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?

MHB Master
#3
January 11th, 2020,
00:55
Thread Author
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

#4
January 13th, 2020,
22:35
Hi Peter,
Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.

MHB Master
#5
January 15th, 2020,
00:02
Thread Author
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

#6
January 16th, 2020,
22:26
Yes, this is correct. Nicely done.

MHB Master
#7
Yesterday,
23:52
Thread Author
Originally Posted by
GJA
Yes, this is correct. Nicely done.
Thanks for all your help, GJA ...
It is much appreciated...
Peter