Inflationary models with a non-zero background curvature are ill-defined in general relativity because scalar modes cannot be canonically quantized. Therefore, there is no consensus on the primordial power spectrum that should be considered at large scales in a curved Universe. In this letter, we propose a model of curved inflation where canonical quantization is possible for any curvature, and we unambiguously obtain the resulting primordial power spectra. The framework is a recently proposed modification of general relativity in which a non-dynamical topological term is added to the Einstein equation. The main strength of this model is that no additional degree of freedom compared to the standard model of cosmology is needed, giving a natural solution to the problem of constructing curved inflation, and at the same time providing an additional argument for this topological modification of general relativity.