Compact subset of a space Consider the topological space $X=\{0,1\}^{\mathbb{Z}}\times \mathbb{Z}$ and let $Y$ be a subset of $X$ given by $Y=\{0,1\}^{\mathbb{Z}}\times\{0,1,2,\cdots\}$. Is $Y$ compact subset of $X$?
Set $U_k=\{0,1\}^{\mathbb{Z}}\times\{k\}$. Then $\{U_k
; k=0,1,2,\cdots\}$ is an open cover of $Y$ having no subcover. So $Y$ cannot be compact. Am I correct?

Yes, you are correct. More generally, if $X$ is any nonempty topological space and $Y$ is infinite discrete, then $X\times Y$ is never compact. That's because $\big\{X\times\{y\}\big\}_{y\in Y}$ is an open cover without finite subcover.
