Is it good to learn measure theory before topology or the other way around?

I think learning topology makes analysis easier to understand: Say, in Euclidean space, a convergent sequence cannot have two distinct limits. In classical mathematical analysis one used to pick ϵ0=|a−b|2ϵ0=|a−b|2 to deduce the contradiction, where a,ba,b are the limit values. But if we switch into a topological point of view, convergent sequence means eventually its elements lie into a neighbourhood of aa, and also in a neighbourhood of bb. Since in Euclidean we can make those two neighbourhoods to be disjoint, the largest radius to the disjoint neighbourhoods would be |a−b|2|a−b|2, that is the reason why we can get a contradiction in assuming the eventually elements lying in those two disjoint neighbourhoods.

It is the Hausdorff property which explains why needs to pick |a−b|2|a−b|2 (or smaller) to be a candidate, this is hard to see if we merely do the usual ϵϵ-argument along with triangle inequalities, it is simply a mess. But with the aid of topology, everything is clear, the strategy makes sense.