Do you think you could help me with some of these? Thank you.
Suppose $A,B⊆X$ are disjoint and $a∈X\setminus B$. Prove the following:
If $B$ is closed, then $d(a,B)>0$
If $B$ is compact then there is some $b∈B$, such that $d(a,B)=d(a,b)$ (so d(a,B)>0)
If $A$ is closed and $B$ is compact, then $d(A,B)>0$
If $A$ and $B$ are compact, then there is some $a$ in $A$ and $b$ in $B$ such that $ d(A,B)=d(a,b)$
Give an example to show that $d(A,B)=0$ is possible for disjoint $A,B⊆X$, with $A$ and $B$ closed.