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theHigherGeometer
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theHigherGeometer
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2026-03-07 23:01:33
raw text
theHigherGeometer Skip to content theHigherGeometer Menu Home About Papers/research Buy me a coffee Small fun observation about finite topological spaces, and some challenges September 20, 2024 ~ David Roberts ~ Leave a comment Here’s a fun thing: if you want to generate a random finite space, instead select a random subset from , the -fold power of the Sierpinski space , since every space embeds into some (arbitrary) product of copies of the Sierpinski space. (Recall that has underlying set , and the only open subsets are . The point is open , which is true for all points in discrete spaces, but not for example in the reals with the standard topology, where they are closed. The point is closed in .) And, in fact, a topological space is if and only if it can be embedded in some power of : in one direction, spaces being closed under arbitrary products and subspaces are standard facts, and then one checks manually the Sierpi...
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