Do open sets stay open?

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Use the definition of an open set to show that if a finite number of points are removed, the remaining set is still open.
Definition:
A set is open if every point of the set lies in an open interval entirely contained in the set.

I'm a bit lost, but I think that I somehow need to show that when a finite number of points are removed, the remaining points are still in the open interval. The problem is that I cannot figure out why removing points would change the location of other points. If I erase points from the open interval, of course the remaining ones would still be in the open interval... right?

The next part of the question reads:
If a countable number of points is removed from an open set the remaining set is not open. To show this, demonstrate a counterexample. Yikes!

I'm really lost on this one. I cannot even see why there would be a difference between removing a countable versus finite number of points.

help...
 

Answers and Replies

  • #2
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Removing points from an open interval makes it so it's no longer an open interval.

First show that the set consisting of the entire space (eg, the entire real line if you are working with the real numbers) minus one point is open. (Hint: is a point open or closed?). Then, use the fact that a finite number of intersections of open sets is an open set. For the second part, think of a sequence of points that converges towards a limit.
 
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