This probably will look like I'm trying to get you to answer my homework, but I'm not. All I'm looking for is to understand the problem and concepts involved with the problem. Here is the problems.
In the following problems in all metric subspaces $X$ of $\Bbb R$ the understood metric on $X$ is the standard metric of $\Bbb R$.
- Let $X = [0,1) ∪ (1,2]$. prove that the subsets $[0,1)$ and $(1,2]$ are both ”clopen” (meaning closed and open at the same time). Hint: use the theorem proved in class characterizing open and closed subsets of subspaces of metric spaces).
Let $E = [0,1)$
Since $E = X\cap(-1,1)$, E is open. Because (-1,1) is open. Thus proving (1,2] is closed because if the E is open, it's compliment is closed.
Now, let $G = (1,2]$
$G = X \cap (1,3)$ Thus proving that G is open, thus proving that E is closed. Both E and G are clopen. QED
- Consider the metric space $\Bbb Z$. Prove that any subset of $\Bbb Z$ is both open and closed in $\Bbb Z$.
Can someone guide me through this?
I know to be a closed set, a set must contain all of it's limit points. To be an open set, a set must contain all it;s interior points. But I can't find the theorem my professor talks about in my real analysis text