(From Pugh's RMA) Let $\mathcal{T}$ be the collection of open subsets of a metric space $M$, and $\mathcal{K}$ be the collection of all closed subsets. Show there is a bijection from $\mathcal{T}$ to $\mathcal{K}$.
I believe the bijection between $\mathcal{T}$ and $\mathcal{K}$ would be the function $f: \mathcal{T} \rightarrow \mathcal{K}$ that returns the interior of each closed subset; the inverse function would return the closure.
I attempted to prove this using the Schroeder-Bernstein Thm: there is a bijection between $A$ and $B$ if there are injective functions $f: A \rightarrow B$ and $g: B \rightarrow A$. So I need to show that the closure and interior functions are injective.
But then I realized the interior function (which takes a closed set $U$ and returns the intersection of all opens contained in $U$) isn't injective-- consider a an interval $[a,b] \in \mathbb{R}$ and consider $[a,b] \cup [p] \in \mathbb{R}$ ($[p]$ is an isolated point). They have the same interiors (I think).
Where does my approach go wrong? Was it a mistake to choose the interior and closure functions to test for injectivity? How would you find the bijection between $\mathcal{T}$ and $\mathcal{K}$?
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