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I'm reading Intro to Topology by Mendelson.

The problem statement is,

Let $A,B$ be nonempty subsets of a topological space $X$. Prove that $A\cup B$ is disconnected if $(\bar{A}\cap B)\cup(A\cap\bar{B})=\emptyset$.

My proof is,

The only way a union of sets are empty is if the individual sets are empty, that is, $\bar{A}\cap B=\emptyset$ and $A\cap\bar{B}=\emptyset$. Yet, we know that $A\subset\bar{A}$ and $B\subset\bar{B}$ and so $A\cap B\subset\bar{A}\cap B$ and $A\cap B\subset A\cap\bar{B}$ and both of the containing sets are empty, which means $A\cap B$ is empty and so $A\cup B$ is a union of disjoint sets and thus disconnected.

My only issue is that $A$ and $B$ are not said to be open, which makes me wonder if my entire approach is wrong. Either way, this is what I could think of.

Thanks for any hints or feedback!

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1 Answer 1

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You’re right to be concerned: it’s not enough to show that $A\cap B=\varnothing$.

Let $C=A\cup B$. Use the fact that $(\operatorname{cl}A)\cap B=\varnothing$ to show that $B$ is a relatively open subset of $C$, and the fact that $A\cap\operatorname{cl}B=\varnothing$ to show ... ?

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I'm assuming what the rest of that sentence is "to show that $A$ is relatively open in $C$ as well." Yet, to begin with, I don't quite see how one would should $B$ is relatively open in $C$. I'm trying to think of what I can intersect with $A\cup B$ which is open and yields $B$, but the only thing that works is to use $B$ itself. –  Shant Danielian Jul 24 '13 at 9:57
    
@Shant: Let $U=X\setminus\operatorname{cl}A$; clearly $U$ is open in $X$, and $U\cap C=B$, so $B$ is relatively open in $C$. –  Brian M. Scott Jul 24 '13 at 10:01
    
I figured out what you were trying to say by your previous comment and was able to follow your lead and show that both $A$ and $B$ were relatively open in $A\cup B$. Now, I've read that a relatively open sets of subspaces aren't necessarily open in the larger topological space and was wondering if this isn't the case here. Since I believe what follows next is that because we found two open sets, which are disjoint, and make up the entire set in their union, means that $A\cup B$ is disconnected. Am I following correctly? –  Shant Danielian Jul 29 '13 at 3:04
    
@Shant: It very likely is the case here, but it’s irrelevant. All that matters in determining that $C$ is not connected is that $A$ and $B$ are open in $C$, not whether they are open in $X$ (which they probably are not). –  Brian M. Scott Jul 29 '13 at 5:44
    
I see. So is the idea that because $C$ is a topological space in its own right, which satisfies the definition of disconnected, that we don't care about whether $A$ or $B$ is open in $X$, only that they're open in $C$? –  Shant Danielian Jul 29 '13 at 8:49

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