Union on the empty set and the set containing the empty set I'm trying to get a clearer sense of some of the consequences the axiom of unions has on the empty set. I understand that $\emptyset = \{\} \not= \{\emptyset\}$. 
But assuming the following identities are correct, I don't understand why $\bigcup\emptyset = \bigcup \{\} = \bigcup \{\emptyset\}$.
It's likely that I'm floundering on some minutiae of set theory, but it's making me uncomfortable, and I'd like to know what I'm missing. 

Mr. Chip already answered your question. Let me add to that in hope of alleviating the discomfort you are experiencing:
We can view $\bigcup$ as a (class) function
$$
\bigcup \colon \mathrm{Set} \to \mathrm{Set}
$$
and viewed as such, your question is in instance witnessing that this function is not injective. There are many such examples, e.g.
$$
\bigcup \{ \{a\}, \{a,b\} \} = \bigcup \{ \{a,b\} \} = \{a, b \}
$$
which may, at first, not be very intuitive but readily follows from the definition. With a little practice, this will become second nature in no time.
