The **down set** (or **downward closed set**) and it's dual **up set** (**upper set** as the book goes by), are the Poset analog of **over**- and **undercategories**.

In this sense, a downset is like the covering space over a Topology, but I'm just parroting the nlab.

[nlab, overcategory](https://ncatlab.org/nlab/show/over+category)

**Puzzle 1 KEP**: Prove that the downset of the top element in a poset \\( P \\) is equivalent to the poset \\( P \\) itself.

**Puzzle 2 KEP**: Prove that the upset of the bottom element in a poset \\( P \\) is equivalent to the poset \\( P \\) itself.

In this sense, a downset is like the covering space over a Topology, but I'm just parroting the nlab.

[nlab, overcategory](https://ncatlab.org/nlab/show/over+category)

**Puzzle 1 KEP**: Prove that the downset of the top element in a poset \\( P \\) is equivalent to the poset \\( P \\) itself.

**Puzzle 2 KEP**: Prove that the upset of the bottom element in a poset \\( P \\) is equivalent to the poset \\( P \\) itself.