Arrow-Debreu Theorem of Existence: Non satiation
Converting my comments into an answer:
At the bottom of p.268, the authors say:

The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is bounded from below.

[Emphasis added.]
Since the Heine-Borel Theorem establishes that $S\subset R^n$ is compact if and only if $S$ is both closed and bounded (from below and above), one cannot conclude necessarily that $X_i$ is compact. (A counterexample is that $[0,\infty)$ is closed but not compact.)