Scott Finnie: thanks for another [nice pictorial explanation!](https://forum.azimuthproject.org/discussion/comment/16571/#Comment_16571)

You wrote:

> Formally, a partition \\(P_1\\) is finer than \\(P_2\\) if:

> 1. \\(P_1\\) and \\(P_2\\) partition the same set, and
> 2. Every part of \\(P_2\\) is a union of parts in \\(P_1\\).

That's right. I find this other equivalent definition to be a bit simpler:

**Definition.** A partition \\(P_1\\) is finer than \\(P_2\\) if:

1. \\(P_1\\) and \\(P_2\\) partition the same set, and
2. Every part of \\(P_1\\) is a subset of some part of \\(P_2\\).

It seems simpler because you just need to take each part of \\(P_1\\) and see if there's _one_ part of \\(P_2\\) that contains it, instead of taking each part of \\(P_2\\) and run around looking for a _bunch_ of parts of \\(P_1\\) whose union gives it.

If it's not obvious that these definitions are equivalent, it's worth pondering them until it _is_ obvious.