Are interior points also limit points

  • Thread starter colstat
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Homework Statement


Are interior points included in (or part of) limit points?


Homework Equations


Since the definition of interior points says that you can find a ball completely contained in the set. For limit points, it's less strict, you just have to find a point other than the center.


The Attempt at a Solution


My guess is yes, but I've never read it anywhere in the book.
 

Answers and Replies

  • #2
You should double check on that but I think the set of all limit points of S is called the closure of S, which is the union of the interior of S and the boundary of S, so the short answer would be: YES
 
  • #3
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You should double check on that but I think the set of all limit points of S is called the closure of S, which is the union of the interior of S and the boundary of S, so the short answer would be: YES

We use open ball to define both limit point and interior point.

Say, we have some set of space, call it S.

When point p is a limit point of S, we say that we can find a point q (q≠p) in B(x, r), meaning an open ball centered at x, with radius r.

When it is interior point, we say all of the points in this ball is also in S.

So, for limit point, you just need one other point. For interior point, you need all points in the ball is also contained in set S.
 

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