Definition of intersection of topologies

I understand that a topology is a collection of subsets of the topological space $X$. Suppose you have topologies $T_1 = \{a_1,a_2,...\}$ and $T_2 = \{b_1,b_2,...\}$ where $a_1,a_2,...,b_1,b_2,...$ are all sets. Am I correct in saying that the intersection of $T_1$ and $T_2$ are only the sets of $T_1,T_2$ that are exactly equivalent? i.e., if $a_1=a_2$. Nothing to do with what these sets cover?

Thus the intersection of $\{ (1,2),(-1,1) \}$ and $\{ (-1,0), (-1,2) \}$ is exactly the null-set? (I know those are not topologies but just for the sake of example)

• Yes, the intersection of topologies is the family of sets that belong to both topologies. – Daniel Fischer Feb 10 '14 at 20:09