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Usually when people want to weaken the union axiom it is to no longer allow infinite unions (such as here), but I want to know is what happens when you weaken it such that open sets only have to give open sets if you can only take the union of sets with non-trivial intersection. Then the axioms would read (adapted from topology without tears):

  1. X and $\emptyset$ are open sets
  2. The union of any (finite or infinite) number of open sets with non-trivial intersection is an open set
  3. The intersection of any finite number of open sets is an open set

So far it seems that for finite sets where you have all the singlets in the topology it no longer forces you to have the discrete topology. Also for finite sets it seems like it is no longer necessarily a lattice under subset as the partial order. Is there anything that breaks with this definition, or is it just uninteresting?

Asaf Karagila
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Jean Du Plessis
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    What's the motivation for this weakening? – Noah Schweber Aug 18 '21 at 21:33
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    Definitions are only interesting if they are useful. So the question is: what's the point? Where would such a structure actually appear? – freakish Aug 18 '21 at 21:47
  • It seems like a natural way to make the topologies on finite sets where all singletons are open not be forced to be the discrete topology. If we also think about structure gluing, there is some notion of in intersection necessary for structures to be glued, so by requiring there to be an intersection you make the open sets glueable – Jean Du Plessis Aug 18 '21 at 22:22

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