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.