Is $M= \{(x,y) \in \mathbb{R^2} : x \in \mathbb{N} \lor y \in \mathbb{N}\} $ polygonally connected?

Question

I'm trying to prove that $M= \{(x,y) \in \mathbb{R^2} : x \in \mathbb{N} \lor y \in \mathbb{N}\} $ is polygonally connected.

I know that polygonally connected implies arc-connected so if I prove that for an $x, y \in M$ there is no arc $\varphi$ joining x and y that verifies $\varphi(t) \in A$ $ \forall t \in [a,b]$ then it won't be polygonally connected, but I'm not quite sure how I could prove it or if I'm taking the wrong path.

Any suggestions?

Answer 1

Let $(x,y)$ and $(x',y')$ be points in $M$ and let $g(x)=\max(0,\lfloor x\rfloor)$. Note that $g(n)=n$ for all $n\in\mathbb N$, $g(x)\in\mathbb N$ for all $x\in\mathbb R$ and hence the following polygonal chain is completely contained in $M$: $$ (x,y) \to (g(x), y) \to (g(x), g(y)) \to(g(x'),g(y)) \\\to (g(x'), g(y')) \to (g(x'), y') \to (x',y'). $$