I'm reading Rourke and Sanderson's Intro to PL topology and I'm having some trouble understanding their definition on polyhedra and since I can't find any other mention of this definition I thought about asking here.
In the book a set $P \subset \Bbb R^n$ is a polyhedron if each point $a \in P$ has a cone neighbourhood with compact base inside $P$, meaning that there is a compact subset $L\subset P$ so that
$$\{\lambda a+\mu b: b\in L,\ \lambda+\mu=1\text{ and } \lambda,\mu\ge0\}\subset P$$
What confuses me is that the empty set is not excluded in the choice of $L$ leading me to think that every singleton in $P$ could be a cone and so every set is a polyhedron. Well that's obviously wrong but I don't seem to understand why.
EDIT. I'm adding the definition of cone, since it's more likely the confusion to be there.
Let $aB$ be the set of all the line segments from $a$ to a point $b\in B$ so that $x = \lambda a + \mu b$ is uniquely expressed or equivalently $a\notin B$ and for different $b_1$, $b_2$ the arcs $[a,b_1]$, $[a,b_2]$ intersect only in $a$
Can you please help me?
They have an ad hoc definition $a\emptyset=\{a\}$ alas. But that does not imply that every set is a polyhedron. What they need for $X$ to be a polyhedron is for each $x\in X$ to have a neighbourhood in $X$ of the form $aL$. But $a\emptyset =\{a\}$ is only open in $X$ iff $a$ is an isolated point of $X$. So their convention does not allow all sets to be polyhedra.