  The influence of the topology and curvature of objects on photonic properties
represents an intriguing fundamental problem from cosmology to nanostructure
physics. The classical topological transition from a ring to a M\"obius strip
is accompanied by a loss of part of the wavelength, compensated by the Berry
phase. In contrast, a strip with the same curvature but without a 180{\deg}
rotation has a zero Berry phase. Here we demonstrate experimentally and
theoretically that the topological transition from a ring to a flat split ring
accumulated both such effects. By cutting a flat dielectric ring of rectangular
cross-section, we observe the lifting of the degeneracy of the CW-CCW modes of
the ring and the formation of two families: topological modes that acquire an
additional phase in the range from 0 to {\pi} depending on the gap width, and
ordinary modes that do not acquire an additional phase. An order parameter is
introduced that accurately describes the magnitude of the spectral splitting of
ordinary and topological modes. We established that an arbitrary non-integer
number of waves can fit along the length of a dielectric split ring resonator,
creating a new avenue in classical and quantum photonic applications.
