I am new to connection issues and I would like to know if you can help me solve the following exercises please, it is not clear to me what relationship you have with connection, thank you.
1.-Let (, ) a metric space, ⊂ a subspace with the restricted metric of . Prove that ⊂ is closed in if and only if it exists ⊂ closed at such that = ∩
2.-Let (, ) a metric space, ⊂ a subspace with the restricted metric of . Prove that if is closed in then ⊂ is closed in if and only if it is closed in
I know this result, but it is that I still do not see topology, I have only seen metric spaces and this comes in the connection part, I would greatly appreciate your help, thank you.
Edit: I have managed to prove the first statement, could you help me with the second one please, or do you think that from the 2nd one you can attack the first one, I would appreciate any advice, please. Thank you.
For the second statement:
$\Rightarrow$) If $A\subset Y$ is closed in $Y$, then $A=F\cap Y$ for some $F\subset X$ closed in $X$, by (1). Since $Y$ is closed in $X$ and the intersection of closed sets in closed, $A$ is closed in $X$.
$\Leftarrow$) If $A\subset Y$ is closed in $X$, then $A=A\cap Y$ is closed in $Y$ again by (1).