Connectedness and path components of the flea-and-comb space

Let \[ X=\{(0,1)\}\cup\{(x,0):0\le x\le1\}\cup \left\{\left(\frac1n,y\right):0\le y\le1,\ n\in\mathbb N\right\} \subset\mathbb R^2 \] with the subspace topology.

1. Show that \(X\setminus\{(0,1)\}\) is path-connected.

2. Show that \(X\) is connected.

3. Show that \(X\) is not path-connected.

4. Prove that \(X\) has exactly two path components, one of which is not closed in \(X\).