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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)

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  • $\begingroup$ Yes, the intersection of topologies is the family of sets that belong to both topologies. $\endgroup$ – Daniel Fischer Feb 10 '14 at 20:09
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The intersection of two sets is the set of objects that are elemetns of boith sets. It doesn't matter indeed tha tthese objects are again sets (that can be intersected, for example). So your example is right.

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