Notation for a collection of sets under a certain condition

Question

I am looking for the notation to describe "A collection of sets that are the union of a finite number of intervals".

Is this correct -

$A = \{A_i\}_{i \in I}$ where each $A_i = \bigcup_{n \in N} U_n$ $\forall \ i$, and each $U_n$ is an interval on $\mathbb{R} \ \forall \ n \in \mathbb{N}$

Answer 1

You should change that to

$A = \{A_i\}_{i \in I}$ where each $A_i = \bigcup_{n =1}^{N_i} U_n$ $\forall \ i$, and each $U_n$ is an interval on $\mathbb{R}$ for $n = 1,\dots,N_i$.

As your statement stands, you imply that each set in your collection is the union of infinitely many intervals.