TITLE: Dense and complete totally ordered sets
QUESTION [5 upvotes]: The totally ordered set $\mathbb{R}$ has the following properties.

[Density]. Given any two $x,y \in \mathbb{R}$, we can find $\eta \in \mathbb{R}$ such that $x < \eta < y$.
[Dedekind Completeness]. Every non-empty subset that admits an upper bound also admits a supremum.

Adjoining a greatest and/or least element to $\mathbb{R}$ preserves the above properties. This gives us a total of four non-isomorphic totally ordered sets satisfying properties 1 and 2, namely $$(-\infty,\infty),\quad[-\infty,\infty),\quad (-\infty,\infty],\quad [-\infty,\infty].$$
This is probably a silly question, but is the above list exhaustive (up to isomorphism)?

REPLY [3 votes]: Additional information for the interested reader:
Joel David Hamkins writes here that

The real line $\langle \mathbb{R},<\rangle$ is (up to isomorphism) the unique nonempty, separable, complete, dense, endless total order.