[:homeomorphism:] [:homeomorphic:] • For a function $f$ between two topological spaces, if $f$ is bijective and both $f$ and $f^{-1}$ are continuous, then we say $f$ is a homeomorphism • Intuition: for $f : X \to Y$, continuity of $f$ means "$f$ may not tear". Continuity of $f^{-1}$ means "$f$ may not glue" • Equivalent to saying: $f(U)$ is open iff $U$ is open • Since $f$ is a bijection, this means that $f$ is also giving us a bijection between open sets • Thus, • "The goal of topology is to classify topological spaces up to homeomorphism" • I.e., the set of spaces modulo being homeomorphic Referenced by: