Weakly contractible

Summary

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

PropertyEdit

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

ExampleEdit

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.

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.

ReferencesEdit

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