are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected. The circle is (strongly) locally contractible but not contractible. Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112). There is some disagreement about which definition is the "standard" definition of local contractibility; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to topological properties. Care should always be taken regarding the definitions when interpreting results about these properties. 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 a standard example of a space which is contractible, but not intuitively so. The Dunce hat is contractible, but not collapsible. 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. The Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not contractible. See also References Topology Homotopy theory Properties of topological spaces