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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.

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.

The reason we can topologize all compact metrizable spaces (up to homeomorphism) is because their cardinality is capped at $\mathfrak c$: π-Base, Search for Compact+Metrizable+~Cardinality $\leq\mathfrak c$. Note also that using the power set trick to produce a missing example fails, since the indiscrete topology is not metrizable for sets with multiple points.

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 missing from the set.

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.

There is no set of all [compact] topological spaces.

Given a set of topological spaces, consider its power set. This power set with the indiscrete topology is a compact topological space missing from the set.

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 missing from the set.