Open Relative to X: Proving T is Open

[ssayan3]

Homework Statement

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

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\capX
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!

 

[HallsofIvy]

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.

 

[ssayan3]

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

 

[ssayan3]

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!