Meaning of the continuous spectrum and the residual spectrum @Konstantin : The continuous spectrum requires that you have an inverse that is unbounded If X X is a complete space, then the inverse cannot be defined on the full space It is standard to require the inverse to be defined on a dense subspace If it is defined on a non-dense subspace, that falls into the miscellaneous category of residual
$f$ is a homeomorphism iff $f$ is bijective, continuous and open 2 Homeomorphism means a continuous bijection whose inverse is continuos too Now use the fact that f is continuous iff for every open set U U of Y , f−1(U) f 1 (U) is open in X The bijection is needed for the other direction, when you have to prove f is homeomorphism f−1 f 1 exists since it is a bijection and continuos as f is an open map