I think this looks good.
You can also show that it is not closed by observing that 0 is the limit of the sequence
Since I've had a marked disability to write proofs I wanted to run this by you guys to insure that it was correct before I moved on.
First I prove that is not open
Since there should exist some interval so because we are using the euclidean topology. Then there should also be or . However, .
Then we need to prove that it is not closed. To do such We prove that the compliment is not open.
. To prove that this is not open we just need to prove that one of the members of the union is not open. Using the same strategy then on let or . Then find the element and see that it is not in . Thus proving that the compliment is not open and that the set is not open nor closed.
Is this correct?
I had to look ahead to see what limit points were but I think I get it.
is a limit point in if every open set contains a point from different from and closed sets contain all their limit points.
So is our open set and it contains points different from that are in , so is a limit point. The interval does not contain so it is not a closed set.
Is this right?
Also since I've now read a bit on the chapter of limit points is say not a limit point of because there's an open interval that contains no points from the set but contains [LaTeX ERROR: Convert failed] ?
I just want to be sure I'm getting the material right and thank you all for all of your help thus far.