Locally contractible space

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This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of contractible space. View other variations of contractible space

Definition

Symbol-free definition

A topological space X is said to be locally contractible if it satisfies the following equivalent conditions:

  1. It has a basis of open subsets each of which is a contractible space under the subspace topology.
  2. For every x \in X and every open subset V \ni x of X, there exists an open subset U \ni x such that U \subseteq V and U is a contractible space in the subspace topology from X.

Formalisms

In terms of the locally operator

This property is obtained by applying the locally operator to the property: contractible space

Note that the locally operator here means the existence of a basis of contractible spaces. It is a stronger condition than merely saying that every point is contained in a contractible open subset; rather, we are claiming that there are arbitrarily small contractible open subsets. The mere condition that every point is contained in a contractible open subset is much weaker.

Relation with other properties

Incomparable properties

  • Contractible space: A contractible space need not be locally contractible; in fact, it need not even be locally connected! An example of a contractible space that is not locally connected is the comb space. Conversely, a locally contractible space need not be contractible. For instance, any manifold is locally contractible, but manifolds such as the circle are not contractible.

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally Euclidean space has a basis comprising subsets homeomorphic to Euclidean space follows from Euclidean spaces being contractible a pair of intersecting lines is locally contractible but not locally Euclidean |FULL LIST, MORE INFO
manifold locally Euclidean of fixed dimension as well as Hausdorff and second-countable (via locally Euclidean) (via locally Euclidean) Locally Euclidean space|FULL LIST, MORE INFO
CW-space underlying topological space (up to homeomorphism) of a CW-complex CW implies locally contractible |FULL LIST, MORE INFO
polyhedron underlying topological space (up to homeomorphism) of the geometric realization of a simplicial complex (via CW-space) CW-space|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally simply connected space has a basis comprising subsets that are simply connected follows from contractible implies simply connected |FULL LIST, MORE INFO
semilocally weakly contractible space |FULL LIST, MORE INFO
semilocally simply connected space |FULL LIST, MORE INFO
locally path-connected space has a basis comprising path-connected subsets |FULL LIST, MORE INFO
locally connected space has a basis comprising connected subsets |FULL LIST, MORE INFO