Show that $((0,1)\times(0,1))\cup(\{1\}\times[0,1])$ is not a smooth manifold with boundary$

Question

I have to show that $\left((0,1)\times(0,1)\right) \cup(\{1\}\times[0,1])$ is not a smooth manifold with boundary. I think that the problem points are (1,0) and (1,1) because their neighborhoods are not diffeomorphic to any open subset of $\mathbb{H}^2$, but I don't know how to show it.

Could anybody help me?

Thanks in advance.