I'm working on topology again and I wanted to know if this proof for a question that I came on is correct.
The question was if the set { } , let's call it , was closed in the Euclidean topology. From prior reading I knew it wasnt since it was missing it's limit point, but I couldn't use that here so I went with this.
Take and suppose that it were open. There are smaller and smaller open intervals beside and not containing smaller and smaller rational numbers approaching zero.
Let be the open interval that surrounds zero. To the right of it there will be an open interval but to the left of that interval will be a point of S no matter how much you shrink .
Therefore the set is not closed.
I understood what he meant, apparently, from the beginning: he needs to show that that set isn't closed
without using at all limits points.
For example, if is closed, then its complement is open, but
this isn't so since , but there is no
open ball around zero which is completely contained in that complement. Q.E.D.
Tonio