Open Relative to X: Proving T is Open

• ssayan3
In summary, if S is open relative to D, and for any point p in S there exists a delta >0 such that B(p,delta) intersects D, then B(p,\delta) is a subset of S.
ssayan3

Homework Statement

T is open relative to X iff for any p $$\in$$ S there exists $$\delta$$ > 0 such that B(p,$$\delta$$ )$$\cap$$X is $$\subset$$ T

Homework Equations

T is open relative to X provided there exists an open subset U of R^n such that T = U$$\cap$$X

The Attempt at a Solution

Okay, so, going the forward ("if") direction, I think I'm able to classify this problem into one of three major subsets: X is closed, open, or neither. So, dealing with X being closed, I so far have:

1. Pick p in T
2. By hypothesis, produce U, an open subset of R^n s.t. T = U$$\cap$$X
3. For all points qn in U, let tau= min{|qn - p|}
4.
...
I believe that the final step will involve producing a delta ball within T, but I dn't know how to go about finding the correct delta. As well, if what I have so far is correct, then I think i'll have to end up letting delta be the minimum of tau and some other variable that I'll call beta for right now; so, how do I find the right beta?

Does anyone have any ideas for the reverse ("only if") direction?

Thanks guys!

Last edited:
I don't see why you would need to think about whether X is open or closed or neither.

If there exist an open subset U of R^n such that $T = U\cap X$, then, since U is open, there exist a ball $B(p,\delta)\subset U$ and so $B(p, \delta)\cap X$ is a subset of T. That's it.

Conceptually, yes, I see what you are talking about. Can you give me another hint to push me along the path of proving it?

Thanks for your help thus far

Haha, well, during the five minutes after I wanted to ask you to give me a hint, I think i pretty much got it figured out...

Forward direction: S is open relative to D if for any point p in S there exists delta >0 s.t. B(p,delta) intersect D is a subset of S

1. Since S is open relative to D, produce U, a subset of R^n, s.t. S = U intersect D
2. B/c U is open, for any point p in U, we can produce delta >0 s.t. B(p,delta) is a subset of U.
3. Pick a point p in U intersect D
4. Then, B(p,delta) is a subset of U intersect D, which is S

Let me know if I forgot any details!

1. What does it mean for a set to be open?

Being open means that every point in the set has a neighborhood that is also contained within the set.

2. How do you prove that a set T is open?

To prove that a set T is open, you must show that for every point in T, there exists a neighborhood of that point that is also contained within T.

3. What is the importance of proving a set is open?

Proving a set is open is important because it allows us to make conclusions about the properties of the set and its elements. It also allows us to use the set in various mathematical operations and proofs.

4. Can a set be both open and closed?

Yes, a set can be both open and closed. This is known as a clopen set. An example of a clopen set is the set of all real numbers between 0 and 1, including 0 and 1.

5. Are there different methods for proving a set is open?

Yes, there are different methods for proving a set is open. Some common methods include using the definition of an open set, proving that the complement of the set is closed, and using topological properties such as continuity or connectedness.

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