what is the relationship of boundary(A intersection B) to boundary(A) and boundary(B)???
What do you think, what does your intuition tell you? What about something like $\displaystyle (a,b),(b,c)$? What is $\displaystyle \partial((a,b)\cap (b,c))$? What about $\displaystyle \partial(a,b)\cap\partial(b,c)$?
,.,.i have this silly intuition.,.,.my intuition says that boundary(A intersection B) is a subset of boundary(A) and boundary(B),.,i know im wrong,.,.hahahah
,.,.i have this silly intuition.,.,.my intuition says that boundary(A intersection B) is a subset of boundary(A) and boundary(B),.,i know im wrong,.,.hahahah
Yes, you are wrong . In $\displaystyle \mathbb{R}$ with the usual topology choose $\displaystyle A=[0,2],\;B=[1,3]$ . Then, $\displaystyle \partial(A\cap B)=\{1,2\}$ and $\displaystyle \partial (A)\cap \partial (B)=\emptyset$ .