# Why are complete lattices non-empty?

I am confused as to whether complete lattices are empty while reading the following article:

Complete Lattice

A complete lattice is a partially ordered set $$(L,\leq)$$ in which every subset of L has a greatest lower bound (glb) and a least upper bound (lub) in L. The lub of the empty set is denoted by $$\bot$$ and glb of the empty set is $$\top$$. It follows that a complete lattice is never empty.

The author just stated that "The lub of the empty set is ... $$\bot$$ and glb of the empty set is $$\top$$.", and the only subset of empty set is itself, so every subset of it does have a lub ($$\bot$$) and glb ($$\top$$), and it should be complete according to this definition. I may be missing something obvious, but

Why is it stated that empty set is not complete here and elsewhere?

-- Edit --

From the answers so far, my understanding is that the statement in question was really stating that the underlying poset of a complete lattice is non-empty.

A further question is, is this non-empty assertion unnecessary?

Since a poset is defined on a set with an ordering, the poset is never empty by definition (e.g. even empty the set has a non-empty powerset). Why is it necessary to assert whether a complete lattice is empty or not (if all posets are non-empty in this sense).

The last two sentences of the quoted passage are, in my opinion, out of order. One should first notice that, in a complete lattice $$L$$, the empty subset must have a supremum and an infimum; these are elements of $$L$$, so $$L$$ isn't empty. Then, introduce the notations $$\bot$$ and $$\top$$ for the supremum and the infimum of $$\varnothing$$ (i.e., the bottom and top elements of $$L$$).
Note that $$\bot$$ and $$\top$$ might be equal (if $$L$$ has only one element), but they (or it) must be present in $$L$$, so $$L$$ can't be empty.
A complete lattice is a partially ordered set $$(L,\leq)$$ in which every subset of $$S$$ of $$L$$ has a greatest lower bound (glb) and a least upper bound (lub) in $$L$$. When $$S=\varnothing$$ the lub of $$S$$ is denoted by $$\bot$$ and glb is denoted by $$\top$$. It follows that a complete lattice is never empty.
Since the empty subset is always a subset, even of the empty set itself, it then has greatest lower and least upper bounds. Therefore, the lattice set $$L$$ contains bounds. It contains something. That means it's not an empty set.