Consider $X = \{ (x, \sin(1/x)) : 0 <x \leq 1 \} \cup \{ (0,y) : -1\leq y \leq 1\}$ as a subspace of $\mathbb{R}^2$ and $Y = [0,1)$ as a subspace of $\mathbb{R}$. Then which of these options are correct?
$X$ is connected.
$X$ is compact.
$X \times Y$ (in product topology) is connected.
$X \times Y$ (in product topology) is compact.
My answer is : option $1)$ and option $3)$ are true: by the graph I can show it's connected.
Option $2)$ and option $4)$ are false because the graph is unbounded so it is not compact.
Is this correct? Any hints/solution will be appreciated.
Thank you.
No! $X$ is compact, The minimum bounded is $1$. For closedness, the second set is the set of limit points of the first one.
For connectedness of $X$, note that the first set, call $A$, in $X$ is the continous image of $(0,1]$ under $x \mapsto \sin 1/x$ . Now $(0,1]$ is connected , so is its continuous image $A$. Also $A$ is connected implies $\overline{A}$ is conneceted. But $\overline{A}=X$ and so $X$ is connected !
The latter two is easy to determine and it is left as an exercise to you :-)