Homeomorphism - Examples

Examples

  • The unit 2-disc D2 and the unit square in R2 are homeomorphic.
  • The open interval (a, b) is homeomorphic to the real numbers R for any a < b. (In this case, a bicontinuous forward mapping is given by f = 1/(x − a) + 1/(x − b) while another such mapping is given by a scaled and translated version of the tan function).
  • The product space S1 × S1 and the two-dimensional torus are homeomorphic.
  • Every uniform isomorphism and isometric isomorphism is a homeomorphism.
  • The 2-sphere with a single point removed is homeomorphic to the set of all points in R2 (a 2-dimensional plane).
  • Let A be a commutative ring with unity and let S be a multiplicative subset of A. Then Spec(AS) is homeomorphic to {p ∈ Spec(A) : p ∩ S = ∅}.
  • Rm and Rn are not homeomorphic for m ≠ n.
  • The Euclidean real line is not homeomorphic to the unit circle as a subspace of R2 as the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact.

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