Let $E = ([0, 1] × \{0\}) ∪ (\{0\} × [0, 1])$ be a subset of $\mathbf{R}^2$ endowed with the metric induced from $(\mathbf{R}^2, ||\cdot||_{\infty})$.
-Surely $E=[0,1]\times[0,1]$, that seems logical to me?
What are the open and the closed balls of centre $(1/2,0)$ in this new metric space.
-I would think the open ball is $(0,1)\times [0,1/2]$ and the closed ball is $[0,1]\times [0,1/2]$
Show that the closure of the open ball is a proper subset of the closed ball
-now this is where I am stuck, sure the closed ball and the closure of the open ball are the same here? That would probably imply my first 2 answers are wrong