How do I designate the infimum of the union of an indexed family of sets of real numbers using set theory? I have an arbitrary collection of sets $\{U_i\}$ from the topology $\Omega := {\varnothing, [0, \infty)} \cup \{(a, \infty): a \in \mathbb{R}_{0+}\}$. In the case the arbitrary collection $\{U_i\}$ doesn't have either $\varnothing$ or $[0, \infty)$, the union of all its members $\bigcup U_i$ is $(a^*, \infty)$ where $a^*$ is the least such $a^*$. Now, for the life of me, I have been suffering trying to designate that set and its infimum with set theory. Is its infimum $a^* = \inf \{a_i : (a_i, \infty) \in U_i\}$? I want to say it is identical to $\inf \{x : (x, \infty) \in U_i \}$, but I'm not sure, I'm a bit confused by the notation. The problem (I think?) is that there's no hint that this set should range over all the possible $U_i$ sets (that have been defined $(a, \infty)$ for $a \in \mathbb{R}_{0+}$) which are in the collection $\{U_i\}$ and grab only the value of the lower bound of the interval.
I feel like this designation should mention the indexing set somehow. Maybe  $\inf \{x : (x, \infty) \in U_i,  i \in I \}$? But in that case, don't I have to define the index set? And how would I define the index set of a collection that is arbitrary? Any hints, help, reading suggestions or keywords are highly appreciated.
 A: Something like $$\inf \{x : (x, \infty) \in U_i \}$$ or $$\inf \{a_i : (a_i, \infty) \in U_i\}$$ is incorrect because $U_i$ refers to an interval (which depends on the index $i$), not a set of intervals, so it would not have something like $(x,\infty)$ or $(a_i,\infty)$ as an element.  A correct notation would be $$\inf \{x : (x, \infty) \in \{U_i\} \}$$ where you are instead writing $\{U_i\}$ for the set of all your intervals.  Another way to write it would be to first define $a_i$ to be the lower endpoint of $U_i$ for each $i$ in your index set $I$ (so $U_i=(a_i,\infty)$, and then take $$\inf\{a_i:i\in I\}.$$ (Note that it's better to first define $U_i=(a_i,\infty)$ outside of the set in the infimum, to avoid any confusion about the status of the variable $i$.  For instance, if you write $\inf \{a_i : (a_i, \infty) \in \{U_i\}\}$, in the expression $(a_i, \infty) \in \{U_i\}$ the two occurrences of $i$ actually have totally different meanings: in $\{U_i\}$ it is locally bound by the index set $I$ since $\{U_i\}$ really means $\{U_i:i\in I\}$, but in $(a_i,\infty)$ the entire expression $a_i$ is instead bound as the variable in the outer set definition.)
You ask about how to define the index set, but the index set is already given to you.  If you write $\{U_i\}$, that is an abbreviation for $\{U_i\}_{i\in I}$ (or $\{U_i:i\in I\}$) for some (perhaps unspecified) index set $I$.  So if you start with a family $\{U_i\}$, that by definition means that you already have an index set which you can freely refer to.  Alternatively, though, you don't need to use index notation at all.  You could instead just say you have a set $\mathcal{F}$ which is a subset of $\Omega$, so $\mathcal{F}$ is equal to the set you've been calling $\{U_i\}$ but with no need for any indexing.  You can then write your infimum as $$\inf \{x : (x, \infty) \in \mathcal{F} \}.$$  This is generally what I would recommend: unless there's some particular reason you want an indexing, you should just use a set, not an indexed family.
