Weakly contractible

Short description: Topological space consisting of trivial homotopy groups

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 [math]\displaystyle{ S^\infty }[/math] to be the inductive limit of the spheres [math]\displaystyle{ S^n, n\ge 1 }[/math]. Then this space is weakly contractible. Since [math]\displaystyle{ S^\infty }[/math] 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

• Hazewinkel, Michiel, ed. (2001), "Homotopy type", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=h/h047940