I want to know an easy to justify logical method of finding the topologies or at least to show that there are 26 topologies on the set. Will greatly appreciate your help.
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2What's wrong with just listing them, one by one? – Arthur Nov 11 '19 at 19:01
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Think first about the two subsets that are open in every topology, and think next about a systematic way to organize and count all the ways to have more open sets. – hardmath Nov 11 '19 at 19:07
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Possible duplicate: https://math.stackexchange.com/q/652354/391136 – Randy Marsh Nov 11 '19 at 19:23
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Have you tried counting the possible topologies of a 2-element set? If not, that wouldbe a helpful warmup. (There are only 4.) – MJD Nov 11 '19 at 19:53
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1There are in fact 29 topologies, 9 if we only count non-homeomorphic ones. See here for a list. And I reproduced it on this site too. – Henno Brandsma Nov 11 '19 at 22:57