Why Use Arbitrary Unions and Finite Intersections in Topology? Why the definition of a topological space defined under finite intersection and arbitrary union
What if we change the conditions by arbitrary intersection and finite union?

General topology arose in large part as a generalization of real analysis. On the real line with the usual topology, an arbitrary union or finite intersection of open intervals is open. 
We cannot push to even countably infinite intersections, however, since
$$
\bigcap_{n=1}^\infty \left(-\frac{1}{n}, \frac{1}{n} \right) = \{0\}
$$
gives us an example of a countably infinite collection of open intervals whose intersection is not open.
