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In book Topology by James Munkres the proof of product of compact spaces is compact uses a fact called Tube lemma as follows

Let $X$ and $Y$ be compact space than the product $X\times Y$ is compact.
Proof: Let $\mathcal A$ be an open cover for the product space $X\times Y$; Since $x_0 \times Y$ in $X\times Y$ is compact (as it is homeomorphic to $Y$) the open cover $\mathcal A$ admits a finite subcover say $\{A_i : i=1\dots m\}$ than $N=\bigcup_{i=1}^m A_i$ is an open set containing $x_0 \times Y$ by using tube lemma there exist an open set $W\times Y$ about $x_0 \times Y$ contained in $N$. Now, for each set of form $x\times Y$ there must be $W_x \times Y$ which is covered by finitely many sets of $\mathcal A$, moreover the set $\{W_x:x\in X\}$ forms an open over for $X$ as we know it is compact therefore the set admits a finite subcover for $X$ i.e $\{W_1, W_2,\dots , W_k\}$ hence the union of the sets $W_1\times Y, W_2\times Y, \dots , W_K\times Y$ form a finite subcover for $X\times Y$.

How does the $W_x \times Y$ which is covered by finitely many sets of $\mathcal A$ help us to conclude the final thing as I only see the compactness of X played a vital role for conclusion.

Thank You.

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You use the compactness of $X$ to find that $W_1\times Y, \dots ,W_K\times Y$ covers $X\times Y$ but this is not necessarily a subcover of $\mathcal{A}$. Luckily the tube lemma said that each $W_i\times Y$ could be covered with finitely many sets of $\mathcal{A}$.

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  • $\begingroup$ that means inherently each $W_i\times Y$ is contained in $N$ which in turn is a union of some finite sets of our cover $\mathcal A$; in other words the compactness of $X$ helped us to guarantee that the overall union is finite and subcover of $\mathcal A$ $\endgroup$ – Tutankhamun Apr 25 '17 at 8:49
  • $\begingroup$ Yes and no, for each $W_i\times Y$ there exists an **$N_i$**(Not the same $N$ for all of them) such that $W_i\times Y$ is covered by $N_i$ and $N_i$ is a finite subcover of $\mathcal{A}$. $\endgroup$ – Mathematician 42 Apr 25 '17 at 8:55

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