$A=\{(x,\sin(1/x):x>0\}$ and $B=\{(0,0),(-1,0)\}$ then $A\cup B$ is connected?

Question

$A=\{(x,\sin(1/x)):x>0\}$ and $B=\{(0,0),(-1,0)\}$ then $A\cup B$ is connected?

My Attempt: $X=A\cup B$ is disconnected iff $cl(A)\cap B=\emptyset$ and $A \cap cl(B)=\emptyset$

Here $A\cap cl(B)=\emptyset$ but $cl(A)\cap B=\{(0,0)\}\neq \emptyset \implies A\cup B$ is not disconnected. So $A\cup B$ is connected.

am I right ?

Answer 1

If $C=\{(x,y): X <-\frac 12 \}$ and $D=\{(x,y): x >-\frac 12 \}$ then $X=(X\cap C) \cup (X \cap D)$. This shows that $X$ is not connected.