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A homeomorphism is a topological isomorphism. More precisely, suppose X and Y are topological spaces, and f is a function from X to Y. Then f is a homeomorphism iff all the following hold:

  1. f is a bijection.
  2. f is continuous.
  3. f -1 is continuous.

If f : X -> Y is a homeomorphism, then Y is said to be homeomorphic to X (or to be a homeomorph of X). If two spaces are homeomorphic then they have exactly the same topological properties.