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Steven Clontz
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There is no topology on the set of all [compact] topological spaces, because there is no set of all [compact] topological spaces.

Given a set of topological spaces, consider the power set of its union. This power set with the indiscrete topology is a compact topological space. It is missing from the original set (even modulo homeomorphism) because it has larger cardinality than every member of the original set.

Steven Clontz
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  • 8
  • 16