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Let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces. Let $f \colon X \rightarrow Y$ be a function. We say that $f$ is a homeomorphism between $(X,\tau)$ and $(Y,\sigma)$ if the following properties are satisfied:

  1. $f$ is a bijection;
  2. $f$ is continuous;
  3. the inverse function function $f^{-1}$ is continuous.