Show that if $X$ is Lindelof and $Y$ is compact, then then $X \times Y$ is Lindelof.
My attempt: let $U=\{ U_\alpha \in \mathcal{T}_p | \alpha \in J\}$ be an open cover of $X \times Y$. Given $x_0 \in X$. Since $\{x_0\} \times Y \cong Y$ and $Y$ Is compact, $\{x_0\} \times Y$ is compact. $U$ is also an open cover of $\{x_0\} \times Y$. So $\exists \{U_{\alpha ,1}, …,U_{\alpha, k}\}$ finite subset of $U$ such that $\{ x_0\} \times Y \subseteq \bigcup_{i=1}^{k} U_{\alpha, i}=N $. So $N\in \mathcal{T}_p$ and $\{x_0\} \times Y \subseteq N$. By lemma 26.8, $\exists W\in \mathcal{N}_{x_0}$ such that $W\times Y \subseteq N$. Thus $\forall x\in X$, $\exists W_x \in \mathcal{N}_x$ such that $W_x \times Y\subseteq N_x =\bigcup \{U_{x,i}|i \in J_{k_x}\}=\bigcup_{i=1}^{k_x}U_{x,i}$. $W’=\{ W_x |x\in X\}$ is an open cover of $X$. Since $X$ is lindelof, $\exists \{ W_{x_n}|n\in \Bbb{N}\}$ countable subcover of $W’$. $\{ W_{x_n}\times Y \in \mathcal{T}_p| n\in \Bbb{N}\}$ is an open cover of $X\times Y$. We need to show $X\times Y =\bigcup_{n\in \Bbb{N}}(W_{x_n} \times Y)$. Inclusion $\supseteq$ holds trivially. Conversely, let $x\times y \in X\times Y$. $\exists m\in \Bbb{N}$ such that $x\in W_{x_m}$. So $x\times y \in W_{x_m} \times Y \subseteq \bigcup_{n\in N}(W_{x_n}\times Y)$. Thus $X\times Y =\bigcup_{n\in \Bbb{N}}(W_{x_n} \times Y)$. We have $W_{x_n}\times Y\subseteq N_{x_n}, \forall n\in \Bbb{N}$. So $X\times Y= \bigcup_{n\in \Bbb{N}}(W_{x_n}\times Y)\subseteq \bigcup_{n\in \Bbb{N}} N_{x_n}$. Hence $X\times Y=\bigcup_{n\in \Bbb{N}}N_{x_n}$. By elementary set theory, $\bigcup_{n\in \Bbb{N}} N_{x_n} =\bigcup_{n\in \Bbb{N}} (\bigcup_{i=1}^{k_{x_n}} U_{x_n,i} )= \bigcup_{n\in \Bbb{N}, i\in J_{k_{x_n}}} U_{x_n,i} $. By theorem 2.13 of Baby Rudin, $\{ U_{x_n ,i} \in \mathcal{T}_p| i\in J_{k_{x_{n}}} ,n\in \Bbb{N}\} \subseteq U$ is countable. So $X \times Y= \bigcup_{n\in \Bbb{N}} N_{x_n} =\bigcup_{n\in \Bbb{N}, i\in J_{k_{x_n}}} U_{x_n ,i}$. Thus $\{ U_{x_n ,i} \in \mathcal{T}_p| i\in J_{k_{x_{n}}} ,n\in \Bbb{N}\}$ is countable subcover of $U$. Is this proof correct? especially notation part.
This post is little bit different than Prob. 14, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: If $X$ is Lindelof and $Y$ is compact, then $X\times Y$ is Lindelof post. Proof of this exercise is very similar to theorem 26.7.