# Thread: Connected Sets in the Plane which are not Disjoint

1. ## Connected Sets in the Plane which are not Disjoint

Let A and B be Connected Sets in the Plane which are not Disjoint. is A intersection B necessarily connected?? What about A union B???

2. ## Re: Connected Sets in the Plane which are not Disjoint

Take $\displaystyle A =\left\{(x,y)\in\mathbb R^2, 0\leq x\leq 1,0\leq y\leq 1\right\}$ and $\displaystyle B=\left\{(x,x),0\leq x\leq 1\right\}$. We have $\displaystyle A\cap B =\left\{(0,0);(1,1)\right\}$.
For the second question, this lemma may help you: A topological space is connected if an only if every continuous map $\displaystyle f:X\rightarrow \left\{0,1\right\}$ is constant.

3. ## Re: Connected Sets in the Plane which are not Disjoint

.,.thnx sir,.,ammm,.,.i've read something like this " a topological space X is said to be connected if there are no proper closed subsets A and B such that A intersection B = null and A union B = X..
so i have somehow concluded that the first question is not necessarily connected. and the second is it is connected,.,.are my conclusions right sir??

4. ## Re: Connected Sets in the Plane which are not Disjoint

Your conclusions are right, now you have to show it.
Did you try to prove the lemma?