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    $\begingroup$ Further thoughts on pointfree topology as an alternative may be found in this MO answer. $\endgroup$ Commented Jul 10, 2025 at 12:10
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    $\begingroup$ This is a really interesting suggestion! I'm curious if anyone knows, historically, how it came to be that open-set-based topological spaces dominated. Was it entirely an accident, or were there reasons that at least seemed good to people at the time? $\endgroup$ Commented Jul 10, 2025 at 16:44
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    $\begingroup$ @Mike Shulman: For how the open set formulation of topological spaces came to dominate, see The emergence of open sets, closed sets, and limit points in analysis and topology by Gregory H. Moore (2008; especially Section 14, but note all the prior formulations discussed, not all of which by the way are actual "logical equivalences" of what is now called a topological space) and the mathoverflow question Why is a topology made up of 'open' sets? $\endgroup$ Commented Jul 10, 2025 at 17:18
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    $\begingroup$ +1, since this also answers “how would you like mathematics to be different?” $\endgroup$ Commented Jul 12, 2025 at 13:56
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    $\begingroup$ Obsessing over the "right" notion of topological space is pointless since (almost) no one actually studies topological spaces. Rather, they study much more geometrically rich and structured objects that happen to be topological spaces, which gives a convenient minimal language to talk about continuity and related notions. You have to choose some such language to get going, but once you start doing real work the point-set details become almost irrelevant. I guess for the purposes of communication it's nice that everyone uses the same formalism. $\endgroup$ Commented Jul 14, 2025 at 19:15