Contractible Space - Examples and Counterexamples

Examples and Counterexamples

  • Any Euclidean space is contractible, as is any star domain on a Euclidean space.
  • The Whitehead manifold is contractible.
  • Spheres of any finite dimension are not contractible.
  • The unit sphere in an infinite-dimensional Hilbert space is contractible.
  • The house with two rooms is standard example of a space which is contractible, but not intuitively so.
  • Dunce hat
  • The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected.
  • All manifolds and CW complexes are locally contractible, but in general not contractible.

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