Implications of the difference between $\sigma$ algebra and topology? The definitions of $\sigma$ algebra and topology on a set are very similar.
Basically the way I think about it is that the topology on $\mathbb R$ contains all the open intervals but not the closed ones, and the sigma algebra contains the closed intervals.
My question is: what is the significance of this difference? What are the main implications of this difference? (I.e. the results that hold for topologies but not for sigma algebras).
 A: A topology on a set is a qualitative way of saying which of its points are close. A $\sigma$-algebra specifies subsets whose volume you can measure. The Borel $\sigma$-algebra on a topological space generally contains many, many more sets than just the open and closed ones (see Borel Hierarchy).
An instance of a result which holds for $\sigma$-algebras but not for topologies is that infinite $\sigma$ algebras are necessarily uncountable.
I think much more interesting differences come to light when the arrows are involved - continuous maps vs measurable maps. In essence, continuous maps preserve shape in the sense of not "tearing" the space, while measurable maps merely preserve measurability (not necessarily measure!) of subsets, and may obliterate shape completely. Specifically, consider the "tearing in half" function $[- 1,1]\to [-2,-1)\amalg [1,2]$ defined by translating $[-1,0)$ to $[-2,-1)$ and translating $[0,1]$ to $[1,2]$. It is discontinuous with respect to the usual topologies (continuity preserves connectedness) but measurable with respect to the usual Borel $\sigma$-algebras. This particular function is also measure preserving since it's a piecewise rigid motion.
Finally, some strong results in measure theory imply that "the structure of a measure space is the coarsest among all the substantial structures on a set" in the sense that the isomorphism class of many topological measurable spaces is determined by their cardinality alone. Consequently many topological measurable spaces cannot admit interesting "geometric" invariants. This is in stark contrast with topological spaces which exhibit invariants such as (co)homology e.g the number of connected components as mentioned in this answer.
