In dedekind's cut the right set considered as open or closed in occasions where the left set is always open. but why this is the only way? won't it be a dedekind's cut if I keep the left set open or closed in occasions and the right set always open?
2026-04-11 14:10:31.1775916631
on dedekind's cut
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It's a convention.
You can define your own "cut", say "KK cut", in such a way that the right set is always open, as you described.
Then there is an obvious bijection between the set of "Dedekind cuts" and the set of "KK cuts", and everything we do with Dekekind cuts can translate to KK cuts.