So i have been studying topology and when proving that the finite product of compact spaces is going to be compact we have to use the tube Lemma, and we have to prove it. I have a question about the proof :
Well we start by covering $ x \times Y $ with basis elements and then since $Y$ is compact and $x \times Y$ is homeomorphic to $Y$ we can find a finite subcolection. My question why do i need this finite sub collection?? If every basis elements $ U \times V$ is in $N$ their infinite union is still gonna be in $N$, there has to be something that i am missing , i guess what im really asking is why the tube lemma doesn't work if $Y$ isnt compact, so any help is appreciated , Thanks.