I know that a "closed" surface is defined as a compact surface with no boundary. I don't have it clear if they have something to do with having an interior volume (completely enclosed volume). The Klein bottle seems to be a closed surface but it does not have an enclosed volume.
Is there a necessary and sufficient property or classification of surfaces that characterizes the ones that do enclose a volume?
By the way, I'm looking into this because I want to characterize the correct kind of surfaces where Gauss's Law (for electric fields) applies in Physics.