Inverse problems deal with determining certain causes for a desired or an observed effect. Naturally they are ill-posed; small errors in the measurement data can lead to poor reconstruction in the solution. This leads us to using regularization methods in approximating the solution. Here, we focus on the convergence of some well-studied regularization methods under general topological spaces. In this framework, the measure of data misfit is not necessarily a norm (in Banach spaces), hence more general assumptions must be employed to prove convergence. We briefly discuss some misfit functionals of such type, which can be found across sciences.