Show a subset is not path connected.

Question

Show that the subset $\{(x, y) \in \mathbb R^2 \mid x\neq 0\}$ of $\mathbb R^2$ is not path-connected.

I know that $X$ is path connected if any two points in $X$ are connected by a path in $X$, but unsure how to show this for this question. Could it been done by contradiction?

Answer 1

The space $A:=\{(x,y) \in \Bbb R^2:x\not=0\}$ is not connected as $U:=\{(x,y) \in \Bbb R^2:x>0\}$ and $V:=\{(x,y) \in \Bbb R^2:x<0\}$ are two disjoint non-empty subset of $A$ with $A=U\sqcup V$. Hence $A$ is also not pathwise-connected.

Answer 2

Let $A$ the set in question. Then $(-1,0), (1,0) \in A.$ Let $c=(c_1,c_2): [0,1] \to \mathbb R^2$ be a continuous path with $c(0)=(-1,0)$ and $c(1)=(1,0).$

Then: $c_1(0)=-1$ and $c_1(1)=1.$ Thus there is $\xi \in (0,1)$ such that $c_1(\xi)=0.$ But then we have that $c( \xi) \notin A.$