Using clopen definition of connectedness to prove $M$ is not connected

Question

Definition: $M$ is disconnected if it contains a proper clopen subset $A$

clopen: closed and open

proper: set is not empty or the entire set

Let $M = \{(x,y) \in \mathbb{R}^2||y|<|x|\}$

Note: origin is missing

Set $M$ looks like:

I am trying to prove this set is disconnected using the above definition but I am trying to produce a clopen set in $M$

Suppose $A$ = Left half plane $\cap M$

$A$ is open but not closed since it does not contain the boundaries as well as the limit point at the origin i.e. a sequence $\{(1/n,0)\}$ fails to converge. Is there a way to make $A$ closed and open? If not, what would be a proper clopen set for this set?

Answer 1

$A= M\cap \{(x,y)\in R^2, x<0\}=M\cap \{(x,y)\in R^2, x\leq 0\}$ so is clopen for the topology of $M$. Remark that a subset of $M$ is open if it is the intersection between $M$ and an open subset of $R^2$ and a subset of $M$ is closed if it is the intersection between $M$ and a closed subset of $R^2$.