  Perovskite structure is one of the five symmetry families suitable for
exhibiting topological insulator phase. However, none of the halides and oxides
stabilizing in this structure exhibit the same. Through density functional
calculations on cubic perovskites (CsSnX$_3$ ; X = Cl, Br, and I), we predict a
band insulator -- Dirac semimetal -- topological insulator phase transition
with uniform compression. With the aid of a Slater-Koster tight binding
Hamiltonian, we show that, apart from the valence electron count, the band
topology of these perovksites is determined by five parameters involving
electron hopping among the Sn-{s, p} orbitals. These parameters monotonically
increase with pressure to gradually transform the positive band gap to a
negative one and thereby enable the quantum phase transition. The universality
of the mechanism of phase transition is established by examining the band
topology of Bi based oxide perovskites. Dynamical stability of the halides
against pressure strengthens the experimental relevance.
