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Suppose that X is a normal topological space, and that $F_1$, $F_2$ are disjoint closed subsets in X. Prove that there exist open subsets $W_1$$F_1$, $W_2$$F_2$, whose closures are disjoint. Here, normal is defined as: A space is normal if for any pair of disjoint closed subsets $F_1$, $F_2$, there exist disjoint open subsets $U_1$$F_1$, $U_2$$F_2$.

The usual approach I take with these sort of problems is to break it down; by picking say a point x in $F_1$ and applying the definition to that, with hope of taking a union over $F_1$. However here I don't know if the singleton {x} is closed so have gotten myself pretty stuck. Looking for a hint.

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Combine two ways of formulating normality:

$X$ is normal (in the sense of separating two disjoint closed sets by disjoint open sets) iff for all closed sets $F$ and open sets $U$ such that $F \subseteq U$ we have an open subset $V$ such that $F \subseteq V \subseteq \overline{V} \subseteq U$.

Proof: This is quite easy to see: suppose $F$ and $U$ with $F \subseteq U$ are given, then $F$ and $U^\complement$ are disjoint and closed so we have disjoint open sets $O_1$ and $O_2$ such that $F \subseteq O_1$ and $U^\complement \subseteq O_2$. But then $O_1 \subseteq O_2^\complement$ by disjointness of $O_1$ and $O_2$ and also $O_2^\complement \subseteq U$; as $O_2^ \complement$ is closed and contains $O_1$, $$F \subseteq O_1 \subseteq \overline{O_1} \subseteq O_2^ \complement \subseteq U$$ and we can take $V=O_1$.

For the reverse, of we have this property and two disjoint closed sets $F_1$ and $F_2$, we have $F_2^ \complement$ open and containing $F_1$ so we have $V$ open with $$F_1 \subseteq V \subseteq \overline{V} \subseteq F_2^\complement$$ and then $O_1= V$ and $O_2=\overline{V}^ \complement$ are the required disjoint open sets separating $F_1$ and $F_2$. QED.


Knowing this, it's easy: separate $F_1$ and $F_2$ by disjoint open sets $O_1$ and $O_2$ resp. and then apply the alternative formulations again for both $F_i$ and $O_i$ ($i=1,2$) to get open neighbourhoods $U_i$ of $F_i$ with closure inside the $O_i$, so a fortiori disjoint too.

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  • $\begingroup$ Thanks for your reply! I'm struggling to see how $O_2^C ⊆ U$ follows though $\endgroup$ – beelal May 18 at 1:20
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    $\begingroup$ @beelal It's equivalent to $U^\complement \subseteq O_2$. Taking complements on both sides reverses inclusions: $A \subseteq B \leftrightarrow B^\complement \subseteq A^\complement$ $\endgroup$ – Henno Brandsma May 18 at 6:45

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