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The physical state of a quantum spin system on an infinite lattice is mathematically interpreted as a state on a quasi-local C*-algebra. For systems with a unique gapped ground state, the ground state is pure and varying the system typically results in a weak*-continuous deformation of the ground state. Inspired by these considerations and by recent works on parametrized topological phases, we study the homotopy theory of the weak* topology on the pure state space in a general C*-algebraic context. Using the selection theory of Ernest Michael, we provide mild hypotheses under which a weak*-continuous family of pure states can be deformed via a homotopy of unitaries so that at the end of the homotopy, the entire family of states evaluates to one on a given projection of the C*-algebra. These deformations can be iterated to show that, in the weak* topology, the pure state space of the quasi-local algebra has trivial homotopy groups of all orders.