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Find all the limit points and interior points (basic topology)

  • Thread starter nalkapo
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  • #1
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Homework Statement



Find all the limit points and interior points of following sets in R2
A={(x,y): 0<=x<=1, 0<=y<=1} *here I used "<=" symbol to name as "less then or equal".
B={1-1/n: n=1,2,3,...}

Homework Equations





The Attempt at a Solution


the limit point of B is 1 as n goes to infinity. because for the limit point, in the neighborhood there must be infinitely many points.
and I think the limit points of A must be all (x,y) values, because in the neighborhood ve can find infinitely many points in the neighborhood of all points.
my problem is with interior points.
How can I approach in order to find interior points? and what are the interior points of these sets???
 

Answers and Replies

  • #2
HallsofIvy
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An interior point of a set A is a point, p, such that some neighborhood of p is contained in A. Look at, for example, points like (0, 1/2), or (1, 3/4), or (2/3, 1), etc. which lie on the lines bounding the rectangle. Any neighborhood must have some points on every side of the point and so some on the outside of the rectangle. Those cannot be interior points.

For the second case, you might want to actually write down some of the points:
0, 1/2, 2/3, 3/4, etc. Suppose you drew an interval, say, (1/4, 3/8) around 1/2 or (2/3- 1/24, 2/3+ 1/24) about 2/3? (1/24 is half of 1/12, the distance from 2/3 to 3/4.)
 
  • #3
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An interior point of a set A is a point, p, such that some neighborhood of p is contained in A. Look at, for example, points like (0, 1/2), or (1, 3/4), or (2/3, 1), etc. which lie on the lines bounding the rectangle. Any neighborhood must have some points on every side of the point and so some on the outside of the rectangle. Those cannot be interior points.

For the second case, you might want to actually write down some of the points:
0, 1/2, 2/3, 3/4, etc. Suppose you drew an interval, say, (1/4, 3/8) around 1/2 or (2/3- 1/24, 2/3+ 1/24) about 2/3? (1/24 is half of 1/12, the distance from 2/3 to 3/4.)
So, you mean (for B={1-1/n}) that for some points p, I cannot find a neighborhood which is a subset of B. for example in [1/4, 3/8] we cannot find any neighborhood which belongs to B. so, in B there is no interior points. Is that true?
 

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