I need to show that the intersection of $T$ and $ \bar{I}$ is equal to empty as well. Lets assume that $T⊂R^{n}$ is open and $I⊂R^n$ & $T\cap I=\varnothing$.
I need to show that the intersection of $T$ and $\bar{I}$ (the closure of $I$) is empty as well. 
How to show this? I have seen this question in a book.but I have No idea. I wonder the solution. Help me. Thank you!

Hint:  Since $T$ is open, the complement of $T$ is closed.  So now what happens when you take the closure of $I$?  How could it possibly have non-empty intersection with $T$?
(Note that $I$ is contained in the complement of $T$, since $I\cap T = \emptyset$.)
