Of course there are many answers to your question. The interesting thing to ask is if there is a "best" or "right" answer. In many respects the "correct" topology for the lattice of open sets is the Scott topology. In case X is locally compact, the Scott topology coincides with the compact-open topology of the continuous function space C(X,Sigma), where Sigma is the Sierpinski space (where we identify open sets with their characteristic functions into Sigma).
There are several reasons why the Scott topology is the "right" one. One of them is that the following are equivalent for a space X:
- X is an exponentiable space in the category of topological spaces (YX exists for all Y).
- The exponential SigmaX exists.
- The topology of X is a continuous lattice.
- The lattice of open sets of X equipped with the Scott topology is the exponential SigmaX.
I recommend the following paper by Martin Escardó and Reinhold Heckmann in which they explain many things related to topology of the lattice of open sets (and function spaces in general):
M.H. Escardo and R. Heckmann. Topologies on spaces of continuous functions. Topology Proceedings, volume 26, number 2, pp. 545-564, 2001-2002.
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