- #1
Bashyboy
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Homework Statement
I read somewhere that if ##\{A_i\}## is a collection of subsets in some topological space ##X## that is locally finite, then ##\overline{\bigcup A_i} = \bigcup \overline{A}_i##, but I am having difficulty showing this.
Homework Equations
The Attempt at a Solution
I already know that ##\bigcup \overline{A}_i \subseteq \overline{\bigcup A_i}## holds independently of whether the collection is locally finite, so it suffices to prove the other inclusion. Let ##a \in \overline{\bigcup A_i}##. Then for every open neighborhood ##U_a## containing ##a##, it will intersect ##\bigcup A_i## which means that ##A_i \cap U_a \neq \emptyset## for some ##i##. Since ##\{A_i\}## is locally finite, there exists an open neighborhood ##U_a## that intersects finitely many sets in the collection...
Clearly my goal is to show that ##a \in \bigcup \overline{A}_i##, and this happens if and only if ##a## is the limit point of some ##A_i##, or that every open of ##a## intersects "consistently" intersects some ##A_i##. My problem is, I don't see how the hypothesis (that ##\{A_i\}## is a locally finite collection) is strong enough to show this happens for all open sets; I can only get that one open neighborhood intersects some of the sets in ##\{A_i\}##, but that's far from showing all open neighborhoods intersect a given ##A_i##. I have thought about this for some time; I could use a few hints.
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