Product of a compact topological space and a singleton in another topological space is compact proof

Question

It's before we prove that 'Product of two compact sets is compact'. There are topological spaces $X$(which is compact), $Y$ and the product topology on $X \times Y$ is given by the subbase $U \times V$ where $U$ is open in $X$ and $V$ is open in $Y$. Pick an element $\bullet \in Y$. Let $S$ be an open cover of $X \times \{\bullet\}$. Then $\pi_1(S)$ is an open cover of $X$ so there is a finite subcover then pick one $A_n$ from $S$ corresponding to each $\pi_1(A_n)$ but I think it may not cover $X \times \{\bullet\}$. How to complete the proof? For example let $Y$ be a $T_1$ space and pick another element $\bullet\bullet \in Y$. There exists an open set $W$ containing $\bullet\bullet$ but not $\bullet$. Lets construct an open cover of $X \times \{\bullet\}$ $$S \cup \{B \times W \mid B \in \pi_1(S)\}$$ Now we can apply this open cover to upper proof, and when we are picking $A_n$ from $S \cup \{B \times W \mid B \in \pi_1(S)\}$ corresponding to each $\pi_1(A_n)$, we may pick all the sets from $\{B \times W \mid B \in \pi_1(S)\}$ so in fact it doesn't cover $X \times \{\bullet\}$. Am I misunderstanding something?

Answer 1

There are two possible interpretations of the sentence "Let $S$ be an open cover of $X \times \{\bullet\}$":

1. Let $S$ be a collection of open subsets of $X \times Y$ which cover $X \times \{\bullet\}$. In this case, you must be careful how you proceed, for the reason that you have observed.

2. Let $S$ be a collection of open subsets of $X \times \{\bullet\}$ which cover $X \times \{\bullet\}$, where $X \times \{\bullet\}$ is given the subspace topology of $X \times Y$. In this case, you can simply observe that $\pi_1: X \times \{\bullet\} \to X$ is a homeomorphism, and the proof you've described works perfectly.

In the case that you are using version 1, observe that the intersection of an open subset of $X \times Y$ with $X \times \{\bullet\}$ forms an open subset of $X \times \{\bullet\}$, which projects under $\pi_1$. onto an open subset of $X$.