Prove that every contractible space is path-connected. Caution: \(X\) being contractible does not imply that \(X\) strong deformation retracts to a point.
Solution
Since \(X\) is contractible, there are \(x_0\in X\) and a homotopy \(H\colon X\times[0,1]\to X\) with \(H(x,0)=x\) and \(H(x,1)=x_0\).
1. For every \(x\in X\), the path \(t\mapsto H(x,t)\) runs from \(x\) to \(x_0\).
Its endpoints are \(H(x,0)=x\) and \(H(x,1)=x_0\).
2. Q.E.D.
For \(x,y\in X\), the path of step 1 from \(x\) to \(x_0\) followed by the reverse of the path from \(y\) to \(x_0\) joins \(x\) to \(y\).