Let $T$ be the collection of open subsets of a metric space $M$, and $K$ the collection of closed subsets. Show that there is bijection from $T$ onto $K$.
Here, I was thinking that
if $f: T\rightarrow K $ is a function such that it maps any open set $U$ to its complement $U^c$, then we are going somewhere. Problem is, I can't figure out how to quite write it out.
Also, how do I show that it is injective and surjective?
If we take two open sets $U_1$ and $U_2 \in$ T such that $U_1 \neq U_2$, then how do I show that the function $f$ maps these two open sets such that $f(U_1) \neq f(U_2)$?
As far as surjection goes, I am lost how to write it out.
Any help will be appreciated. thanks