How to draw Hasse's diagram for an infinite poset (real numbers)?

Question

How can I depict a poset of an infinite set like the real numbers using a Hasse's diagram ? Clarification: The order is the usual order of the infinite set.

Answer 1

The Hasse diagram of a poset $P$ is the directed graph with vertices $P$ and an edge from $a$ to $b$ whenever $a<b$ and there exists no $c$ such that $a<c<b$. In the case $P=\mathbb{R}$, there are no pairs $(a,b)$ with this property, so the Hasse diagram has no edges. Of course, this does not actually tell you anything about the order of $\mathbb{R}$. In general, the Hasse diagram of an infinite poset typically does not contain enough information to recover the poset.