[: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
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