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?

While defining topology in general, mathematicians wanted to generalize the  understanding of "openess" from metric spaces such as $\mathbb{R}^N$, defined by the means of open balls. 
Thus, for example, the intersection of sets of the form $(-\frac{1}{n},\frac{1}{n})$ (where $n$ is any integer) should not be open, since it equals a singleton point $0,$ which is not open in the metric topology. 
Similarly, the union of sets of the form $(-r,r)$ (where $r$ is any real number) should be open, since the union equals $\mathbb{R}$, which is open in metric topology.
