We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction ($x$) of otherwise randomly selected {\it disconnected} sites of a parent crystal, feature a quantized global topological invariant ({\it strong} PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical $x$ below which such systems foster a {\it fragile} PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any $x$, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these {\it possibly} generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via $x$, is characterized by the {\it mean} correlation length exponent $ν\in (1.00,1.46)$.
