I am reading the proof of that the product of finitely many compact spaces is compact from munkres their is a certain little step I don't understand 100 %.
The way he proved it is that he divided the proof into two steps:
Step 1:
Let N is an open set around the slice $x_0 \times Y$ he proved that their is a tube i.e open set W in X that contains $x_0$ such that N contains $W\times Y$.
I understand the proof for step 1, however their is little part I don't understand in step 2.
Step 2:
Let X and Y be compact spaces. Let $A$ be a open covering of $X\times Y$. Given $x_0$, the slice $x_0 \times Y$ is compact since it is homeomorphic to Y, so it can be covered by finitely many elements $A_1,…,A_m$ of A. Their union $N = A_1 \cup … \cup A_m$ is an open set containing $x_0 \times Y$. Here is my question why does it contain $x_0 \times Y$ shouldn't their union be equal to $x_0 \times Y$ by the definition of compactness ? because this will make a huge difference in the proof in the other parts.
Best Answer
For a compact space you are right: The famous finite subcover is a collection of open subsets of the space such that theri union equals the space. Here we talk about a compact subspace, so we have to deal with relative topology, or relatively open sets. By definition of that, the $A_i\cap (\{x_0\}\times Y)$ are open in the space $\{x_0\}\times Y$ (or which is almost the same, in $Y$). That the union of finitely many $A_i\cap (\{x_0\}\times Y)$ equals $\{x_0\}\times Y$ is just another way of saying that the union of the $A_i$ contains $\{x_0\}\times Y$.