TITLE: Is the empty set (or any analogy) ever non-unique?
QUESTION [5 upvotes]: I seems SO obvious that the empty set is unique, and yet every lecture on set theory and topology feels the need to state the proof. Why is that? 
Usually if something seems completely obvious, but the lecturer still presents the proof, it is a tell-tail sign, that one is studying the special case without knowing, and the property isn't true in the general case.

REPLY [2 votes]: The question seems to ask about motivations of people which is not a mathematical question. However, consider a set theory which allows urelements (atoms), that is, objects which have no elements, yet are not the empty set. The standard proof of uniqueness of empty set does not apply to atoms. Also, consider the category of sets and functions. The empty set is an initial object in this category. Any initial object in any category is unique up to isomorphism. This is similar to the uniqueness of empty set in set theory.