Notation for defining cover of a set Is it appropriate to define a cover $C$ of non-singleton subsets of $S$ using the following notation?
$$
C := \left\{ X \subseteq S: \vert X \vert \ge 2
\  \text{and} \ 
\bigcup_{X \in C} X = S
\right\},
$$
where $\vert \cdot  \vert$ denotes the cardinality of a set.
In particular, I am annoyed by the presence of $C$ in the left- as well as in the right-hand side.
 A: When using set-builder notation, the first half of the notation specifies a dummy variable used to denote elements of the set, and the second half of the notation specifies properties which elements of the set must have.  In the notation
$$C := \left\{ X \subseteq S: \vert X \vert \ge 2
\  \text{and} \ 
\bigcup_{X \in C} X = S
\right\},$$
the expression $\bigcup_{X\in C} X = S$ does not specify properties of the elements $X$, but rather attempts to specify a property of the set $C$ itself.  This is bound to cause problems.  Even if we eliminate this expression, I don't think that the notation expresses what you want.  Specifically, the expression
$$
C := \left\{ X \subseteq S: \vert X \vert \ge 2\right\}
$$
is the collection of all non-empty, non-singleton subsets of $S$.  This happens to be a cover, but is a very specific cover.  If it is what you are actually meaning to express, then the union is redundant.  Otherwise, the notation is not the notation you want.
Instead, I would write something like the following:

Let $S$ be a set.  A cover of $S$ is a collection $\mathscr{C} \in \mathscr{P}(X)$ such that
$$ S \subseteq \bigcup_{C \in \mathscr{C}} C. $$
Fix a cover $\mathscr{C}$ of $S$ such that $C \in \mathscr{C}$ implies that $|C| \ge 2$.

A: Using $s$ for both the dummy variable in the definition of the set and the dummy variable in the union is indeed a problem. The union itself is also problematic: $S$ is $\{s:s\in S\}$, not $\bigcup\{s:s\in S\}$. Finally, even if the definition said what I think you intend, it wouldn’t define anything: there is nothing in it that picks out any specific cover of $S$. (That does change if you simply delete the problematic union: then $C$ is defined to be the collection of all subsets of $S$ having at least two elements.)
If you want to define a cover of $S$, you must actually specify how its members are chosen. If you simply want to say that $C$ is a cover of $S$ with certain properties, you can do that, but you won’t be defining $C$. For instance, you can say ‘Let $C$ be a cover of $S$ such that $|c|\ge 2$ for each $c\in C$.’
