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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...