Proving Openness of Set S in R^2

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In summary, to prove that the following set is open, you must find a distance r such that all points within r of (x,y) are also in S.
  • #1
poutsos.A
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How do we prove that the following set is open??


S= [tex]R^2[/tex]- {(x,y) : y=[tex]x^2[/tex], xεR}

WHERE R is for real Nos
 
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  • #2
How about using the definition of "open set", which, in a metric space, is that every point of the set is an interior point. Let (x,y) be a point in S which means it is a pair of real numbers such that y is NOT equal to [itex]x^2[/itex]. Then show that there exist a distance r such that all points within r of (x,y) are also in S.

One way to do that is to find the shortest distance from (x,y) to the curve [itex]y= x^2[/itex] and take half that distance as r.

Frankly, it is hard to see how you could be expected to do a problem like this if you could not see how to prove that the limit of a constant sequence is that constant.
 
  • #3
Thanx, but how do we calculate that distance d(x,y) =r??
 
  • #4
poutsos.A said:
Thanx, but how do we calculate that distance d(x,y) =r??

distance formula?
 
  • #5
poutsos.A said:
Thanx, but how do we calculate that distance d(x,y) =r??
As long as you know there is such a non-zero distance, you don't need to know what it is!
 
  • #6
From the definition of the open set we have:

S is open iff for all xεS THERE exists r>0 such that Disc(x,r) [tex]\subseteq S[/tex]

........or(equivalently)..........


S is open iff for all xεS ,there exists r>0 such that, yεDisc(x,r) =====> yεS.


Let now x=(k,l) ,y=(m,n) then yεDisc(x,r) <=====>[tex]\sqrt{(k-m)^2+(l-n)^2}[/tex]< r

AND the above becomes:



S is open iff [if (k,l)εS then there exists r>0 such that ,if [tex]\sqrt{(k-m)^2+(l-n)^2}[/tex]< r then (m,n)εS].


But (k,l)εS <====> k< [tex]\l^2[/tex], (m,n)εS <====> m< [tex]\ n^2[/tex]. And if w=(o,p) is a point on the curve THEN the distance between y and w is :

....[tex]\sqrt{(m-o)^2 + (n-p)^2}[/tex]..........

AND if we choose r< [tex]\sqrt{(m-o)^2 + (n-p)^2}[/tex]...the above becomes:



S is open iff .

......if... k< [tex]\l^2[/tex] and o=p^2 and [tex]\sqrt{(k-m)^2+(l-n)^2}[/tex]< [tex]\sqrt{(m-o)^2 + (n-p)^2}[/tex]... then m< [tex]\ n^2[/tex]

Can anybody curry on from here?

Also note k< [tex]\l^2[/tex] is one case another is k> [tex]\l^2[/tex]
 

1. What does it mean to prove the openness of a set in R^2?

To prove the openness of a set in R^2 means to show that the set contains all of its interior points. In other words, every point in the set has a neighborhood contained entirely within the set.

2. How do you prove the openness of a set in R^2?

To prove the openness of a set in R^2, you can use the definition of openness, which states that a set is open if every point in the set has a neighborhood contained within the set. This can be shown by using the distance formula to find the distance between a point in the set and all other points in the set, and then finding a radius that is smaller than the distance to ensure that all points within that radius are also in the set.

3. What is the importance of proving the openness of a set in R^2?

Proving the openness of a set in R^2 is important because it allows us to determine whether a set is continuous or not. In other words, if a set is open, we can be sure that it does not contain any boundary points, which are points that are both in and not in the set. This is a crucial concept in various areas of mathematics, including calculus and topology.

4. Can a set be both open and closed in R^2?

No, a set cannot be both open and closed in R^2. A set is considered open if it contains all of its interior points, while a set is closed if it contains all of its boundary points. Since a set cannot contain both interior and boundary points, it cannot be both open and closed.

5. Are there any alternative methods for proving the openness of a set in R^2?

Yes, there are alternative methods for proving the openness of a set in R^2, such as using the definition of openness in terms of limit points or using the concept of open balls. These methods can be useful in different scenarios and can provide alternative perspectives on the openness of a set.

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