I don't know any useful applications of proving that the union is associative when used with indexed families of sets and I have a hard time visualizing and having intuition on why the union is associative when dealing with union of indexed families of sets. I do understand that the union is associative for non-indexed families of sets.
I think that the associativity of the union on indexed families of sets has no applications and isn't useful and everything it does can be expressed with a regular non indexed family of sets because of ignorance which I invite you to correct.
I'm confused on why we need to index S and I using the indexed set I and delta (the triangle). If i$\in$ $I_\lambda$ and i$\in$ I, is either $I_\lambda$ or I a subset of one another respectively because this means $I_\lambda$ $\cap$ I $\neq$ $\varnothing$ because $\exists$ i $\in$ $I_\lambda$ $\cap$ I?
Besides that, I don't understand why union is associative with family of sets. I'm looking for intuition or visualization.