  Disorder and localization have dramatic influence on the topological
properties of a quantum system. While strong disorder can close the band gap
thus depriving topological materials of topological features, disorder may also
induce topology from trivial band structures, wherein topological invariants
are shared by completely localized states in real space. Here we experimentally
investigate a fundamentally distinct scenario where a topological phase is
identified in a critically localized regime, with eigenstates neither fully
extended nor completely localized. Adopting the technique of momentum-lattice
engineering for ultracold atoms, we implement a one-dimensional, generalized
Aubry-Andr\'e model with off-diagonal quasi-periodic disorder in momentum
space, and characterize its localization and topological properties through
dynamic observables. We then demonstrate the impact of interactions on the
critically localized topological state, as a first experimental endeavour
toward the clarification of many-body critical phase, the critical analogue of
the many-body localized state.
