  Distinct from the dynamical phase, in a cyclic evolution, a system's state
may acquire an additional component, a.k.a. geometric phase. The latter is a
manifestation of a closed path in state space. Geometric phases underlie
various physical phenomena, notably the emergence of topological invariants of
many-body states. Recently it has been demonstrated that geometric phases can
be induced by a sequence of generalized measurements implemented on a single
qubit. Furthermore, it has been predicted that such geometric phases may
exhibit a topological transition as function of the measurement strength. Here,
we demonstrate and study this transition experimentally employing an optical
platform. We show the robustness to certain generalizations of the original
protocol, as well as to certain types of imperfections. Our protocol can be
interpreted in terms of environment-induced geometric phases.
