In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 is said to be weakly contractible if all of its homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

s are trivial.

Property

It follows from Whitehead's Theorem

Whitehead theorem

In homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....

 that if a CW-complex is weakly contractible then it is contractible.

Example

Define to be the inductive limit of the spheres . Then this space is weakly contractible. Since is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more.