It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic problems using their representation in the correspondent topological space.

Natural duality theory provides a general framework that encompasses many important and well-known representation theorems in algebra. In fact, when dualities are useful, the duals are simpler structures: the dual of a free algebra is a direct exponent.

In Pontryagin duality, the structure of the dual space is the same when adding the topology. As the semilattice duality Hofmann-Mislove-Starlka, the Pontryagin duality is for algebras for which the basic operations are homomorphisms. The same happens with vector spaces.

Gelfand Duality is a representation theorem for commutative Banach algebras describing them as algebras of continuous functions. Moreover, for commutative $C^{*}$-algebras, this representation is an isometric isomorphism. It is also a classical example of the contributions of lattice duality to topology.

A famous example is the proof of Tychonoff theorem without using the axiom of choice: if you pass from topological spaces over to the algebraic side and use locales, then one can show that a product of compact locales is compact, without choice (cf. Peter Johnstone's book Stone spaces).

Otherwise, the main use of lattice dualities seems to be rather one-sided, i.e., (citing "Natural Dualities for the Working Algebraist") to translate algebraic problems, usually stated in an abstract symbolic language, into dual, topological problems, where geometric intuition comes to our help.

Are there other clear examples of problems of topological nature solved by lattice duality? Thank you in advance.