In rereading Thurston's essay *On Proof and Progress in Mathematics* I ran across this passage:

… this means that some concepts that I use freely and naturally in my personal thinking are foreign to most mathematicians I talk to. My personal mental models and structures are similar in character to the kinds of models groups of mathematicians share—but they are often different models. At the time of the formulation of the geometrization conjecture, my understanding of hyperbolic geometry was a good example. A random continuing example is an understanding of finite topological spaces, an oddball topic that can lend good insight to a variety of questions but that is generally not worth developing in any one case because there are standard circumlocutions that avoid it.

When reading his papers one very quickly sees what he means about how unique his way of thinking about hyperbolic geometry was. I'm more intrigued by the remark about finite topological spaces: I'm aware they can yield quick counterexamples to a few naïve conjectures in general/algebraic topology. They are also clearly related to the theory of lattices, which are extremely useful.

But aside from the Sierpiński space (whose importance I appreciate the reasons for more easily) and the examples above how have finite topological spaces been used to yield insights into other parts of mathematics? And what would be some instances where ‘standard circumlocutions’ were used to avoid them?

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