I have started working through Rourke and Sanderson's ``Introduction to Piecewise-Linear Topology" and have come to a snag quite early on. Among the first few examples of polyhedra are the following,
$\mathbb{R}^{n}$ is a polyhedron. The subset $\mathbb{R}_{+}^{n}\subset\mathbb{R}^{n}$, defined by $x_{n}\geq 0$, is a polyhedron. Subspaces of $\mathbb{R}^{n}$ are polyhedra.
It may also be worth mentioning that the metric used on $\mathbb{R}^{n}$ is the the sup metric.
The first two of these I was able to work out alright. This last one about subspaces is tricky. In part because I am unfamiliar with the authors' definition of subspace, which is the following:
``Linear" always means linear in the affine sense; thus a linear subspace (or just a subspace) $V\subset\mathbb{R}^{n}$ is a translated vector subspace, or equivalently: for each finite set $\{a_{i}\}\subset V$ and real numbers $\lambda_{i}$, with $\sum\lambda_{i}=1$ we have $\sum\lambda_{i}a_{i}\in V$.
Polyhedra are defined in the following way:
Let $A,B\subset\mathbb{R}^{n}$. Define their join $AB$ to be the subset $AB=\{\lambda a+\mu b|a\in A< b\in B\}$ where $\lambda,\mu\in\mathbb{R}$, $\lambda,\mu\geq 0$ and $\lambda+\mu=1$. If $A=\emptyset$ we define $AB=B$. If $A=\{a\}$ is a singleton we denote the join $\{a\}B$ by $aB$. We say that $aB$ is cone with vertex $a$ and base $B$ if each point not equal to $a$ is expressed uniquely as $\lambda a+\mu b$ where $b\in B$, $\lambda,\mu\geq 0$, $\lambda+\mu=1$.
A subset $P\subset\mathbb{R}^{n}$ is a polyhedron if each point $a\in P$ has a cone neighbourhood $N=aL$ in $P$, where $L$ is compact. Note that the case $L=\emptyset$ is not excluded so that a point is a polyhedron.
By playing with some examples I've come to find that these subspaces of $\mathbb{R}^{n}$ will look like copies of some $\mathbb{R}^{m\leq n}$ that are, as stated, translated. In such a case one could work out the above example by simply saying that subspaces are homeomorphic to some $\mathbb{R}^{m}$.
Is my interpretation accurate? Also, what would be a good reference for reading about spaces that are linear ``in the affine sense"? Thanks.