Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post).
Although topology is usually motivated as a study of spatial structures, you can interpret topological spaces as being a particular type of logic, and give a purely logical, non-spatial interpretation to a number of bits of topology.
This seems like one of those facts that was obvious to everyone else already, but I’ll write a quick blog post about it anyway.
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