Weakly contractible

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

Then this space is weakly contractible.

See Contractibility of unit sphere in Hilbert space for more.

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.