Let $X$ be the set of integers with the topology generated by sets of the form $a + k\mathbb{Z} = \{a+k\lambda|\lambda \in \mathbb{Z}$}, where $a,k \in X$ and $k \neq 0$. The basis sets are sipmly the cosets of nonzero subgoups of the integers.
This is from the book "Counterexamples in Topology" by Steen/Seebach. It goes on to say that points in $X$ are closed but not open, and i'm having trouble making sense of that.
I would like somebody to explain to me the two different reasons why each point is closed, i.e. I'm wondering:
How is the complement of a point a union of basis elements?
How do i know that a set containing just a single point contains all its limit points?
Thanks everyone, I have a feeling this is going to be one of those "oh duh" questions once somebody points out to me the reasoning.
Best Answer
The set $\{0\}$ is closed because $\mathbb{Z}\setminus\{0\}$ is the union of the following open sets:
A similar argument applies to any other integer.
In order to see that if $n\neq0$ then $n$ is not a limit point of $\{0\}$, just note that there are always integers $a$ and $k$, with $k\neq0$, such that $0\notin a+k\mathbb Z$ and that $n\in a+k\mathbb Z$; just take $a=n$ and $k=2n$, for instance. So, $n\in a+k\mathbb Z$ which is an open set to which $0$ does not bolong. Therefore, $n$ is not a limit point of $\{0\}$.