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

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In summary, Peter is trying to use $N_{\epsilon}(p)\subset O$ to prove that $N_{\epsilon}(p)\cap X$ is a set inequality.
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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:
View attachment 9514Can 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
 

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  • #2
Hi Peter,

Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?
 
  • #3
GJA said:
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
Hi Peter,

Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.
 
  • #5
GJA said:
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
Yes, this is correct. Nicely done.
 
  • #7
GJA said:
Yes, this is correct. Nicely done.
Thanks for all your help, GJA ...

It is much appreciated...

Peter
 

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

1. What is the significance of "Relatively Open Sets" in Stoll's Theorem 3.1.16 (a)?

The concept of "relatively open sets" is important in topology, as it allows for a more flexible definition of open sets. In Stoll's Theorem 3.1.16 (a), it is used to prove that a continuous function from a compact space to a Hausdorff space is closed. This has applications in many areas of mathematics, including analysis and geometry.

2. How does Stoll's Theorem 3.1.16 (a) relate to other theorems in topology?

Stoll's Theorem 3.1.16 (a) is closely related to the more well-known Tietze Extension Theorem, which states that any continuous function defined on a closed subset of a normal space can be extended to a continuous function on the entire space. In fact, Stoll's theorem can be seen as a generalization of the Tietze Extension Theorem.

3. Can you give an example of a "relatively open set" in topology?

One example of a relatively open set is the set of all real numbers greater than or equal to 0, which is relatively open in the set of all real numbers. This means that it is open when considered as a subset of the non-negative real numbers, but not necessarily open in the larger set of all real numbers.

4. How does Stoll's Theorem 3.1.16 (a) contribute to our understanding of compact spaces and Hausdorff spaces?

Stoll's Theorem 3.1.16 (a) shows that the combination of a compact space and a Hausdorff space has certain desirable properties, such as being closed under continuous functions. This is important in topology, as compact and Hausdorff spaces are often used to prove more general theorems.

5. Are there any applications of Stoll's Theorem 3.1.16 (a) outside of mathematics?

While Stoll's Theorem 3.1.16 (a) is primarily used in mathematics, it has applications in other fields as well. For example, it can be used in physics to prove the existence of solutions to certain differential equations, and in computer science to analyze the behavior of algorithms. It also has applications in economics and game theory, where it can be used to model decision-making processes.

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