I'm reading Kollar's paper: Toward moduli of singular varieties Compositio Mathematica, tome 56, no3 (1985), p. 369-398. http://archive.numdam.org/ARCHIVE/CM/CM_1985__56_3/CM_1985__56_3_369_0/CM_1985__56_3_369_0.pdf
Corollary 3.4.5 is left unproved, which should be standard and trivial to experts. But I can not figure it out. Can anybody give a sketch how it works?
As a consequence of Theorem 3.4.4 in that paper, the corollary says that minimality is an open condition. Theorem 3.4.4 says that if $f:X\rightarrow T$ is a flat deformation of a projective variety $X_{0}$ over the spectrum of a DVR, and if $X_{0}$ has only minimal singularities, then the general fibre $X_{t}$ has only minimal singularities as well. If I'm correct, Corollary 3.4.5 asserts that if we replace $T$ above by a scheme $S$ (maybe higher dimension), we have an open subsheme $S'\subset S$ such that fibers over $S'$ have the same property with $X_{0}$.
I found an old post in the Mathoverflow: https://mathoverflow.net/questions/82183/well-known-facts-on-openness-condition, which asked a similar question, but without any answer. I would also like to know the answerer.
Thanks!