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If we define a topology via open sets then yes, infinite unions of the finite subsets will necessitate the infinite subsets also part of the topology giving us the discrete space; however, if we define via closed sets then I was thinking only finite unions of closed sets need be in the topology and since finite unions of finite sets are finite, then aren't the infinite subsets still left out of this topology? Therefor, it's possible to define a topology over an infinite set via closed sets that includes all finite subsets but does not include any infinite subsets (Besides the entire set)--then, since all subsets are not in this topology, this would not be the discrete topology.

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Correct.

What you describe is the cofinite topology: the open sets of which are exactly the complements of finite subsets and the whole space.

When the base set is infinite, it's indeed not the discrete topology; it's not even Haussdorf.

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