Proving B(r,x) is a Subset of S^c: Basic Topology Question

[trap101]

Assume ##|X| > \rho## , let ##r = |X| - \rho##

Now I am trying to show that ##B(r,x)\subseteq S^c##

This should be a simple question, but I am struggling trying to find the right inequlity.

Attempt:

let ##y## be a point in ##B(r,x)##.

I know that ##|x - y| < r##.

I have to somehow show that ##|y| > \rho##

this is where my argument falls apart:

##|y| \leq |y-x| + |x|< r + \rho## (by triangle inequality)

but this doesn't show that ##|y| > \rho##

what am I missing?

 

[HallsofIvy]

It's awfully hard to figure out what you are trying to say here.

Assume |X| > [itex]\rho[/itex] , let r = |X| - [itex]\rho[/itex]

Is "X" a number and |X| its absolute value (as opposed to a vector and its length)? If a vector or point, in R2, R³, or Rⁿ?

Now I am trying to show that B(r, x) [itex]\subset[/itex] S^c

Okay, Is "x" the same as "X", above? I presume that B(r, x) is the ball of radius r centered on x (in some metric space) but what is S^c? I would guess "the compliment of set S" but what is S?

This should be a simple question, but I am struggling trying to find the right inequlity.

Attempt:

let y be a point in B(r,x).

I know that |x - y| < r.

I have to somehow show that |y| > [itex]\rho[/itex]

this is where my argument falls apart:

|y| <= |y-x| + |x| (by triangle inequality) < r + [itex]\rho[/itex]

but this doesn't show that |y| > [itex]\rho[/itex]

what am I missing?

 

[trap101]

It's awfully hard to figure out what you are trying to say here.


Is "X" a number and |X| its absolute value (as opposed to a vector and its length)? If a vector or point, in R<sup>2, R3, or Rn?


Okay, Is "x" the same as "X", above? I presume that B(r, x) is the ball of radius r centered on x (in some metric space) but what is Sc? I would guess "the compliment of set S" but what is S?</sup>

^{I apologize my writing of the question is very messy. You are right on both counts. X is a vector in Rn, Sc is the complement of S, and S is a subset of Rn. Sorry for making it messy.}

 

[HallsofIvy]

Then your question makes no sense. You are given that [itex]|x|> \rho[/itex], and you want to prove that [itex]B(r, x)\subset S^c[/itex], the complement of set S? But what is "S". You don't mention it in the hypotheses.

 

[trap101]

Then your question makes no sense. You are given that [itex]|x|> \rho[/itex], and you want to prove that [itex]B(r, x)\subset S^c[/itex], the complement of set S? But what is "S". You don't mention it in the hypotheses.

Sorry. S is B([itex]\rho[/itex], 0) , that is the ball of radius [itex]\rho[/itex] about the origin.

 

[Fredrik]

OK, you want to show that if ##0<\rho<\|x\|## and ##r=\|x\|-\rho##, then ##B(x,r)\subseteq B(0,\rho)^c##. Let ##y\in B(x,r)## be arbitrary. You want to show that ##\|y\|>\rho##. I suggest this as the first step:
$$\|y\|\geq\|x\|-\|y-x\|.$$

 

[trap101]

OK, you want to show that if ##0<\rho<\|x\|## and ##r=\|x\|-\rho##, then ##B(x,r)\subseteq B(0,\rho)^c##. Let ##y\in B(x,r)## be arbitrary. You want to show that ##\|y\|>\rho##. I suggest this as the first step:
$$\|y\|\geq\|x\|-\|y-x\|.$$

Ok using: $$\|y\|\geq\|x\|-\|y-x\|.$$

I was trying this before but didn't feel it would be valid. Now since ##r=\|x\|-\rho##, I can then say

## r + \rho = \|x\|##

now can I say that:

##\|y\|\geq\|x\|-\|y-x\|\geq r + \rho = \|x\| - r##

which wold reduce to ##\|y\|\geq\rho ##

 

[Fredrik]

Ok using: $$\|y\|\geq\|x\|-\|y-x\|.$$

I was trying this before but didn't feel it would be valid.

Are you familiar with the triangle inequality in the form ##\|x+y\|\geq \|x\|-\|y\|##? (This can be derived from the usual version). Write ##\|y\|=\|(y-x)+x\|## and then use this.

## r + \rho = \|x\|##

now can I say that:

##\|y\|\geq\|x\|-\|y-x\|\geq r + \rho = \|x\| - r##

This should be ##\|y\|\geq\|x\|-\|y-x\| > r+\rho-r##. (Because ##\|x\|=r+\rho## and ##\|y-x\|<r##).

 

[trap101]

This should be ##\|y\|\geq\|x\|-\|y-x\| > r+\rho-r##. (Because ##\|x\|=r+\rho## and ##\|y-x\|<r##).

Yea I just copied the code wrong for that, but no i was not aware of the triangle inequality in that form. I'm gping to go derive it now. Thanks. If I have an issue I will ask for assistence