# Is My Proof of Theorem 3.1.16 from Stoll's Real Analysis Book Correct?

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In summary, the proof shows that a subset U of X is open in X if and only if it can be written as the intersection of X with some open subset O of the real numbers. This result is useful in many areas of real analysis.
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I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need further 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 9515
Can someone please help me to demonstrate a formal and rigorous proof of the following:If the subset $$\displaystyle U$$ of $$\displaystyle X$$ is open in $$\displaystyle X$$ ...

... then ...

$$\displaystyle U = X \cap O$$ for some open subset $$\displaystyle O$$ of $$\displaystyle \mathbb{R}$$ ...
Help will be much appreciated ...
My attempt at a proof is as follows:Assume that $$\displaystyle U$$ is open in $$\displaystyle X$$.

Then for every $$\displaystyle p \in U \ \exists \ \epsilon \gt 0$$ such that $$\displaystyle N_{ \epsilon_p } (p) \cap X \subset U$$ ...

Then the set $$\displaystyle \cup \{ N_{ \epsilon_p } (p) \ : \ p \in U \}$$ is open ... since it is the union of open sets ...

Now we claim that $$\displaystyle \cup \{ N_{ \epsilon_p } (p) \ : \ p \in U \} \cap X = U$$ as required ...Proof that $$\displaystyle \cup \{ N_{ \epsilon_p } (p) \ : \ p \in U \} \cap X = U$$ proceeds as follows:Let $$\displaystyle x \in \cup \{ N_{ \epsilon_p } (p) \ : \ p \in U \} \cap X$$

Then $$\displaystyle x \in U$$ and $$\displaystyle x \in X$$ ... so obviously $$\displaystyle x \in U$$ ... ...
Let $$\displaystyle x \in U$$

Then $$\displaystyle x \in N_{ \epsilon_x } (x) \cap X$$

Therefore $$\displaystyle x \in \cup \{ N_{ \epsilon_p } (p) \ : \ p \in U \} \cap X$$

Is the above proof correct?

Peter

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Your proof looks correct to me. Let me explain the steps in more detail to make it clearer.

First, we know that U is open in X, so for every point p in U, there exists an open ball N_{\epsilon_p}(p) that is contained in U. This is the definition of an open set.

Next, we take the union of all these open balls. This set is also open because it is the union of open sets. This is a key property of open sets.

Now we claim that the intersection of this set with X is equal to U. To prove this, we need to show that any point x in U is also in the intersection, and any point in the intersection is also in U.

For the first part, if x is in U, then by definition, there exists an open ball N_{\epsilon_x}(x) that is contained in U. But since x is also in X, this ball is also contained in X. Therefore, x is in the intersection.

For the second part, if x is in the intersection, then it is in an open ball N_{\epsilon_p}(p) for some point p in U. But since the intersection is with X, this ball must also be contained in X. Therefore, x is in U.

This completes the proof that U is equal to the intersection of the union of open balls with X. This intersection is also an open set, since it is the intersection of open sets.

I hope this helps clarify the proof for you. Let me know if you have any further questions.

## 1. What is the definition of "relatively open sets"?

"Relatively open sets" refer to subsets of a topological space that are open in relation to another subset. In other words, a set A is relatively open in a set B if A can be written as the intersection of B with an open set in the larger topological space.

## 2. How is the concept of "relatively open sets" relevant in mathematics?

The concept of relatively open sets is important in topology and analysis, as it allows for the study of open sets within a larger space. It also allows for the comparison of open sets in different topological spaces.

## 3. Can you provide an example of a relatively open set?

Consider the set A = (0,1) in the real numbers, and the set B = [0,1]. A is relatively open in B, as it can be written as the intersection of B with the open set (-1,2) in the real numbers.

## 4. What does Theorem 3.1.16 (a) in Stoll's book state?

Theorem 3.1.16 (a) in Stoll's book states that if A is a relatively open set in a topological space X, then A is open in X.

## 5. How is Theorem 3.1.16 (a) useful in mathematics?

Theorem 3.1.16 (a) is useful in proving the openness of certain subsets in a topological space. It also helps in understanding the relationship between open sets and relatively open sets in a larger space.

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