Difference between topology and sigma-algebra axioms. One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically but is there a way to think; so that we can define a geometric difference? In other words I want to have an intuitive idea in application of this objects. 

I just wanted to add a remarkable comment on this as I was reading through a great book on probability theory by Achim Klenke on p.8
Differences between topologies and sigma-algebra
In contrast with σ -algebras, topologies are closed under finite intersections only i.e. why the answer of the previous question applies as a case in point. Furthermore, the topologies are also closed under arbitrary unions while σ-algebras need not be closed under arbitrary unions, only under countable unions.
I hope this clarifies the matter
