Isomorphic duality of two lattices

Question

If $\mathcal (X , T)$ is a topological space then the closed subsets as well as the open subsets form a lattice using $\subseteq$ as the partial order. But I need to show that these two lattices are isomorphic dual to each other. What should I prove ? How to proceed.

Answer 1

To show duality between lattices $L$ and $M$, it suffices to exhibit a bijection $f:L\to M$ such that $$\forall l_{1},l_{2}\in L \, \left[l_{1}\leq l_{2} \iff f(l_{1})\geq f(l_{2})\right].$$