# relation of boundary and connected

Let $$X$$ be a connected topological space, $$U$$, $$V\in X$$ two non-disjoint open subsets none of which contains the other one. Prove that if their boundaries $$Fr(U)$$ and $$Fr(V)$$ are connected, then $$Fr(U)\cap Fr(V)\neq\emptyset$$.

I've tried that since $$X$$ is connected, and $$U, V$$ are proper subsets of $$X$$, then $$Fr(U)$$ and $$Fr(V)$$ are not empty. And also since $$Fr(U)$$ and $$Fr(V)$$ are connected, then $$Cl(U)$$ and $$Cl(V)$$ are connected. Then $$Cl(U)\cap V\neq\emptyset$$, and $$Cl(U)\cap (X-V)\neq\emptyset$$, then $$Cl(U)\cap Fr(V)\neq\emptyset$$, but I can not see this goes anywhere, any help, thanks!

• what are $A$ and $B$? – ΑΘΩ Dec 21 '19 at 1:01
• Should be U and V, typed wrong – Cathy Dec 21 '19 at 1:02

After edit: Still false. Consider two points $$x,y\in X=\Bbb R^n$$, and then $$U=X\setminus\{x\}$$ and $$V=X\setminus\{y\}$$.
Before edit: False. Consider $$X=\Bbb R^2$$, $$U=\{(x,y)\in\Bbb R^2\,:\, x^2+y^2<1\}$$ and $$V=\{(x,y)\in\Bbb R^2\,:\, (x-1)^2+y^2<1\}$$.
• typed wrong, sorry, the question should be $Fr(U)\cap Fr(V)\neq\emptyset$ – Cathy Dec 21 '19 at 1:08