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. 

An easy way to get a feeling for this is to consider basic examples.
For example, let $X=\{1, 2, 3\}$.
A topological space $(X, )$ could be for constructed by choosing for example $=\{∅,\{1, 2\},\{2\},\{2,3\},X\}$.
But this is as far from a $σ$-algebra as you can get since in fact no complement of any set in $$ is in  except for $X$ and $∅$.
Have a look at some examples of topologies, some examples of $σ$-algebras and try to compare them. Start easy (like this) and move on to some harder ones and you will develop an intuition after hand.
