How can we prove true that:
If $f$ is proper, then the set $T_d=\{s\in S\mid \dim f^{-1}(s) \geq d\}$ is closed?
Is this true if we do not require $f$ to be proper?
(I think the semicontinuity in the dimension of fibres look like Hartshorne Thm 12.8., but I don't know how to express dimension as the cohomology of certain sheaf if it works)
Are there any references?