# union-of-connected-subsets-is-connected-if-intersection-is-nonempty [duplicate]

Union of connected subsets is connected if intersection is nonempty

I don't understand why A∩F and B∩F are relatively open where Brian Scott commented.

Thanks

## marked as duplicate by Did, Siminore, user99914, Aloizio Macedo♦, drhabOct 4 '15 at 14:40

• $A, B$ are open in $\bigcup F$ and $F \subset \bigcup F$. Thus $A\cap F, B\cap F$ are relatively open in $F$. – user99914 Oct 4 '15 at 9:28
• I believe the OP is talking about the 2 comments after Brian M. Scott's answer. It is not clear to the OP why $A\cap F$ and $B\cap F$ are relatively open in $F$. – R_D Oct 4 '15 at 13:39
$A \cap F$ and $B \cap F$ are open in $F$ by definition of subspace topology on $F$. A subset $X$ of $F$ is open in $F$ iff there exists an open subset $U$ of $M$ such that $X = F \cap U$.
Since $A$ and $B$ are open subsets of $\bigcup \mathscr{F}$ there are open sets $U$ and $V$ in $M$ such that $A=U \bigcap (\bigcup \mathscr{F})$ and $B = V \bigcap (\bigcup \mathscr{F})$. So $A\cap F = U \cap F$ and $B \cap F = V \cap F$, making them open in $F$.