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
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.
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]