Weakly contractible

In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.

Property

It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible.

Example

Define ${\displaystyle S^{\infty }}$ to be the inductive limit of the spheres ${\displaystyle S^{n},n\geq 1}$. Then this space is weakly contractible. Since ${\displaystyle S^{\infty }}$ is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more.

The Long Line is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex. Another prominent example for this phenomenon is the Warsaw circle.

References

• "Homotopy type", Encyclopedia of Mathematics, EMS Press, 2001 [1994]