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We demonstrate that monolithic photonic crystals (PCs) confined to the facet, or tip, of single-mode optical fibers can be designed as highly sensitive refractive index point sensors. Monolithic photonic-crystal slabs are fabricated on silicon wafers and subsequently released and micro-assembled on the tip of optical fibers. The PC slab's reflection spectrum is modulated by the refractive index of the environment and shows sensitivity comparable to fiber Bragg gratings (FBGs). The compact sensing element is located at the tip of a fiber and hence is a highly localized sensor. The sensor is made of robust, high-temperature dielectric materials and therefore has the potential to be used for many applications including measurements in harsh environments.