Consider the subset $A$ and $B$ of $\mathbb{R}^2$ defined by $A =\{(x, x\sin\frac{1}{x}) :x\in(0,1]\}$
$B = A\cup \{(0,0)\}$
I have to check for compactness and connectedness of $A$ and $B$.
Here is my attempt.
$A$ is bounded but not closed as 0 is the limit point of set $A$ but it doesn't belongs to $A$. Hence $A$ is not compact.
$B$ is compact since it is closed and bounded subset of $\mathbb{R}^2$.
I am not able to figure out connectedness of given sets.
Am I correct? Is there any other way to tackle this problem? I need help with this.
Thank you very much