General Topology – Difference Between Topology and Sigma-Algebra Axioms

Question

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.

Answer 1

I would like to mention that in An Epsilon of Room, remark 1.1.3, Tao states:

The notion of a measurable space (X, S) (and of a measurable function) is superficially similar to that of a topological space (X, F) (and of a continuous function); the topology F contains ∅ and X just as the σ-algebra S does, but is now closed under arbitrary unions and finite intersections, rather than countable unions, countable intersections, and complements. The two categories are linked to each other by the Borel algebra construction.

Later, in example 1.1.5:

given any collection F of sets on X we can define the σ-algebra B [ F ] generated by F , defined to be the intersection of all the σ-algebras containing F , or equivalently the coarsest algebra for which all sets in F are measurable. (This intersection is non-vacuous, since it will always involve the discrete σ-algebra 2^X). In particular, the open sets F of a topological space ( X, F ) generate a σ-algebra, known as the Borel σ-algebra of that space.