  Recent theoretical studies predict that structural disorder, serving as a
bridge connecting a crystalline material to an amorphous material, can induce a
topological insulator from a trivial phase. However, to experimentally observe
such a topological phase transition is very challenging due to the difficulty
in controlling structural disorder in a quantum material. Given experimental
realization of randomly positioned Rydberg atoms, such a system is naturally
suited to studying structural disorder induced topological phase transitions
and topological amorphous phases. Motivated by the development, we study
topological phases in an experimentally accessible one-dimensional amorphous
Rydberg atom chain with random atom configurations. In the single-particle
level, we find symmetry-protected topological amorphous insulators and a
structural disorder induced topological phase transition, indicating that
Rydberg atoms provide an ideal platform to experimentally observe the
phenomenon using state-of-the-art technologies. Furthermore, we predict the
existence of a gapless symmetry-protected topological phase of interacting
bosons in the experimentally accessible system. The resultant many-body
topological amorphous phase is characterized by a $\mathbb{Z}_2$ invariant.
