What is the partition of an empty set?

Definition 11.1.10 (Partitions). Let I be a bounded interval. A partition of $I$ is a finite set $P$ of bounded intervals contained in $I$, such that every $x$ in $I$ lies in exactly one of the bounded intervals $J$ in $P$.

It is trivial to prove an empty set is the partition of an empty set. However, I found a set whose only element is an empty set could be also a partition of an empty set. Here is my proof, based on the definition of a partition in Analysis I of Tao.

Assume P is not a partition of an empty set, then P could contain some set that is not contained in an empty set. It means an empty set is not a subset of itself, which in turn means that there is some element in an empty set that is not in itself. However an empty set contains nothing, so an empty set must be a subset of itself. Now, the only other case is that there exists some element in the empty set such that it is either not found in any element of P or belongs to more than one elements of P. But again an empty set does not contain anything, so that proves P which contains an empty set is a partition of that empty set.

Could you help to point out if I am correct?

Thanks

1 Answer

One option is to decide that the empty set is not a bounded interval, so this definition does not apply to it!

If we pretend that the empty set is a bounded interval and apply the definition of partition to the empty set, then indeed, both $\emptyset$ and $\{\emptyset\}$ would be partitions, although your proof of the latter case is unnecessarily convoluted: it is vacuously true that, for all $x \in \emptyset$, there is some $J \in P = \{\emptyset\}$ such that $x \in J$, since there are no elements $x$ to check.

While playing pretend is fun, it is worth noting that this would make the quoted definition of partition inconsistent with other uses of that term; typically the parts in a partition are required to be non-empty.

[Some credit is due to Oscar Cunningham; I wrote this answer after a discussion on the subject held on the Category Theory Zulip forum.]