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Do finite topologies have any practical uses other than for counterexamples

user169197
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    http://mathoverflow.net/questions/177461/how-much-of-homotopy-theory-can-be-done-using-only-finite-topological-spaces – Qiaochu Yuan Aug 11 '14 at 21:14
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    FINITE TOPOLOGIES! HUH (yeah)! What are they good for? ABSOLUTELY NOTHING! (Say it again!) – Asaf Karagila Aug 11 '14 at 21:18
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    @Asaf A one syllable word to a six syllable phrase? That won't do at all! –  Aug 11 '14 at 21:29
  • These answers 1 2 discuss an important application of the Sierpiński space, which is a topological space with two points. (It is the unique topological space with two points that is neither the discrete nor the indiscrete space.) – MJD Aug 11 '14 at 21:30
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    Sometimes I wonder if the question we should really be asking is 'Do we really need infinite topological space when finite space seem to be good enough?' – Dan Rust Aug 12 '14 at 12:03
  • @daniel Infinite topological spaces are models of intuitionistic logic. Finite spaces never are. – MJD Aug 12 '14 at 12:29
  • @MJD Sorry I was being a little tongue-in-cheek. I don't seriously consider finite spaces enough to cover everything we do with infinite spaces, but I think the link I gave highlights just how much homotopy theory we can actually do with just finite spaces. – Dan Rust Aug 12 '14 at 12:34
  • I think this could be a good question if phrased as a reference request. – Austin Mohr Aug 12 '14 at 18:36
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    @AustinMohr You can still give answers supported by references, whether or not references are explicitly asked for... –  Aug 17 '14 at 03:10

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For instance the Sierpinski space $(S, \tau)$ with $S = \{0,1\}$ and $\tau = \{\emptyset, \{0\}, S\}$ is quite useful, see http://en.wikipedia.org/wiki/Sierpi%C5%84ski_space . Also, with this kind of "minimal sandbox" topology you can learn about closures, continuous maps, and so on without having to deal with infinite sets.

Dominic van der Zypen
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