In my homework there is an exercise that asks to show the following result:
Let $(E,d)$ be a metric space. Show that a subset $A$ is dense in $E$ iff every open set in $(E,d)$ contains an element of $A$.
I was thinking in the case of the empty set. My question:
"$\emptyset$ contains an element of $A$" is false or is vacuously true?
If it is false, then the necessary condition for the denseness of $A$ will always be false, because there will always be the (open) empty set in $E$ which does not contain any element of $A$. In this case, logically, $A$ would never be a dense subset of $E$. Is my argument right or am I going crazy?
Thanks in advance.