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

References

Hazewinkel, Michiel, ed. (2001) [1994], "Homotopy type", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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