Prove that a regular Lindelöf topological space is normal

Question

That a closed subset of a Lindelöf space is Lindelöf, is already proven. I use this in my proof of the following:

Prove that a regular Lindelöf topological space is normal.

Here is my proof:

Let $ X $ be a regular Lindelöf topological space and let two disjoint closed sets $ A $ and $ B $ be given. Since $ X $ is regular, for each $ x\in A $ there exists open disjoint sets $ U_x $ and $ V_x $ such that $ x\in U_x $ and $ B\subset V_x $. Let $ \mathcal{U} $ and $ \mathcal{V} $ be the set of all such sets $ U_x $ and $ V_x $, respectively, for every $ x\in A $. The set of sets defined by $ \mathcal{W}=\{U_x\cap A \} $ for all $ x\in A $ is an open covering of $ A $. Since $ A $ is a closed set in a Lindelöf space, $ A $ is Lindelöf as well by the previous problem and there exists a countable subcollection $ \mathcal{A} $ of $ \mathcal{W} $ that covers $ A $. Let $ U_0 $ be the intersection of all sets of $ \mathcal{A} $ and let $ V_0 $ be the intersection of all $ V_x $ corresponding to the sets of $ \mathcal{A} $. Since $ U_x\cap V_x=\emptyset $ for all $ x\in A $, it is clear that $ U_0\cap V_0=\emptyset $. Also, since a countable intersection of open sets is open, $ U_0 $ and $ V_0 $ are both open. Since $ V\subseteq V_x $ for each $ x\in A $, $ V\subseteq V_0 $ as well. In summary, $ U_0 $ and $ V_0 $ are disjoint open sets containing $ A $ and $ B $, respectively, and hence $ X $ is normal.

Question: Is my proof correct? I think it has a couple of mistakes. Can I be sure that every set in $\mathcal{W}$ is open in $A$? And it is simply wrong that a countable intersection of open sets is open, right? How can I correct my proof, and is there a different and better proof?

Answer 1

At line 4 of your proof, "$\mathcal{W}=\{U_x\cap A \}$ for all $x\in A$ is an open covering of $A$" is wrong because $U_x\cap A$ may not be open anymore ($A$ is closed).

The actual proof is much harder and as follows.

Since $X$ is regular, for any $x\in A$, there are open sets $U_x, G_x$ such that $x\in U_x$ and $B\subset G_x, \:U_x\cap G_x=\varnothing$. So $G_x^c\subset B^c$ and $U_x\subset G_x^c$. Since $G_x$ is open, $G_x^c$ is closed. Thus $$ U_x\subset \overline{U_x}\subset \overline{G_x^c}\subset G_x^c\subset B^c $$ Clearly $\mathcal{W}=\{U_x|x\in A\}$ is an open cover of $A$. Since $X$ is a Lindelöf topological space, there is a countable subcover of $\mathcal{W}$ for $A$, i.e. $$ A\subset \bigcup_{n=1}^{\infty} U_n, \quad U_n\cap B=\varnothing\tag1 $$ Likewise, there is a countable open cover for $B$, i.e. $$ B\subset \bigcup_{n=1}^{\infty} V_n, \quad V_n\cap A=\varnothing\tag2 $$ where $V_n$ is open and $V_n\subset \overline{V_n}\subset A^c$. Now let $$ O_n=U_n\cap \bigcap_{i=1}^{n} \overline{V_i}^c\quad\text{and}\quad W_n=V_n\cap \bigcap_{i=1}^{n} \overline{U_i}^c $$ Since $\overline{U_i}, \overline{V_i}$ are closed, $\overline{U_i}^c, \overline{V_i}^c$ are open. So $O_n, W_n$ are open by the fact that a finite intersection of open sets is open. Since for any $n$, $\overline{V_n}\subset A^c$, $A\subset \overline{V_n}^c$. Likewise, $B\subset \overline{U_n}^c$. So $$ A\subset \bigcap_{n=1}^{\infty} \overline{V_n}^c\quad\text{and}\quad B\subset \bigcap_{n=1}^{\infty} \overline{U_n}^c $$ Thus by $(1)$ $$ \bigcup_{n=1}^{\infty}O_n=\bigcup_{n=1}^{\infty} \left(U_n\cap \bigcap_{i=1}^{n} \overline{V_i}^c\right)\supset\bigcup_{n=1}^{\infty} U_n\cap\bigcap_{n=1}^{\infty} \overline{V_n}^c\supset A $$ Likewise, by $(2)$ there is $$ \bigcup_{n=1}^{\infty}W_n\supset\bigcup_{n=1}^{\infty} V_n\cap\bigcap_{n=1}^{\infty} \overline{U_n}^c\supset B $$ So $\bigcup_{n=1}^{\infty}O_n$ and $\bigcup_{n=1}^{\infty}W_n$ are open covers of $A$ and $B$ because arbitrary union of open sets is open. Furthermore, WLOG suppose $n\geqslant m$ $$ O_n\cap W_m=\left(U_n\cap \bigcap_{i=1}^{n} \overline{V_i}^c\right)\cap \left(V_m\cap \bigcap_{i=1}^{m} \overline{U_i}^c\right)\subset \overline{V_m}^c\cap V_m\subset {V_m}^c\cap V_m=\varnothing $$ So $O_n\cap W_m=\varnothing$. Thus $$ \bigcup_{n=1}^{\infty}O_n\cap \bigcup_{n=1}^{\infty}W_n=\bigcup_{n, m=1}^{\infty}(O_n\cap W_m)=\varnothing $$ i.e. $\bigcup_{n=1}^{\infty}O_n$ and $\bigcup_{n=1}^{\infty}W_n$ are disjoint. Hence $X$ is a normal space.

Answer 2

By regularity we can construct $U_x$ for all $x$ in $A$ such that $\lbrace U_x \rbrace$ is an open cover of $A$ and $U_x$ closure is disjoint from $B$. $A$ is Lindelöf as it is closed subspace of Lindelöf space $X$. So, the cover $\lbrace U_x \rbrace$ has a countable subcover $\lbrace U_n \rbrace$. Now similarly we can construct a countable open cover $\lbrace V_n\rbrace$ Of $B$ such that $U_n$ closure disjoint from $A$. Now we can proceed as we have proof in Munkres 32.1(normal space) for "regular space with countable basis is normal".