I'm at the end of my first course on general topology, but this topic was not well developed. I can tell that an homeomorphism preserves the quality of a point to be a boundary point for a subset of a topological space. In particular, from space X to space Y, one only needs a function to be countinuos (or something else? I think there's not even need for bijectivness).
But what happens when we talk about The boundary as a whole? What happens in terms of connected component et cetera?
Am I right to say we can distinguish two subspaces by their boundaries even when this is not included in the subspace? (Specially when considering the topology on an open subset).