title: Weakly contractible
text: # 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.
  
  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.
  
  