As the topic, suppose $A,B$ are connected set with $A\cap \bar{B}\ne \emptyset$, does it implies $A\cup B$ is connected?
$A,B$ are connected set with $A\cap \bar{B}\ne \emptyset\implies A\cup B$ is connected??
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general-topology
1 Answers
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If $A\cup B$ is not connected but $A$ and $B$ are, the only possible separation of $A\cup B$ is into components $A$ and $B$. But if $\{A,B\}$ is a separation of $A\cup B$, there must be open sets $U$ and $V$ such that $U\cap(A\cup B)=A$ and $V\cap(A\cup B)=B$. Clearly, then, $A\subseteq U\subseteq X\setminus B$, where $X$ is the whole space; is this possible if $A\cap\operatorname{cl}B\ne\varnothing$?
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0@Mathematics: Yes, I do. – 2012-11-21