Show that if there is a path between two points, they lie in the same connected component. Why does this show that a path connected space is connected?
Solution
Suppose \(x,y\in X\) are joined by a path \(\gamma:[0,1]\to X\). The interval \([0,1]\) is connected, so its continuous image \[ \gamma([0,1]) \] is connected. This connected subset contains both \(x\) and \(y\). Therefore \(x\) and \(y\) lie in the same connected component of \(X\), since a connected component is the maximal connected subset containing either point.
If \(X\) is path connected, fix \(x_0\in X\). Every \(x\in X\) is connected to \(x_0\) by a path, hence lies in the same connected component as \(x_0\). Thus that component is all of \(X\), so \(X\) is connected.