(Sorry for the slanted image.) This is from Munkre's pg10. I do not understand what is meant by
Being closed and discrete...
What I deduce from the arugment is (i) $B' \subseteq B$ is closed in $|K|$, hence in $B$. (ii) Any one point set in $B$ is open in $B$ (complement of closed set.) (iii) So $\{x\} \subseteq B$ satisfies, for some open set $O$ in $A$ , $O \cap B = \{x\}$. (iv) This contradicts $A$ being limit point compact.
Is this what is meant?
By definition, a space is discrete if all its subsets are open; taking complements, this is equivalent to saying that all its subsets are closed. The author notes that every subset of $B$ is closed (in $|K|$) by definition of the polytope topology. By definition of the subspace topology (intersect a closed set in $|K|$ with $B$), this means that every subset of $B$ is closed in $B$, i.e., that $B$ is discrete. Therefore $B$ is both discrete and closed in $|K|$ (hence also in $A$, since $B = B \cap A$).