Arrow-Debreu Theorem of Existence: Non satiation

Question

Let $n$ be the number of consumers and $m$ be the number of commodities.

The Arrow-Debreu theorem requires closed and convex consumption sets $X_i \subset \mathbb{R}^m$ for all buyers $i \in [n]$. Additionally, it requires the utility function of any consumer $i$, $u_i: \mathbb{R}^m \to \mathbb{R}$ to be continuous, quasi-concave and non-satiated over the consumption set $X_i$, where non satitation is defined as $\forall \mathbf{x} \in X_i, \exists \mathbf{y} \in X_i$ such that $u_i(\mathbf{y}) > u_i(\mathbf{x})$ (pages 268-269).

It seems to me like these assumptions are contradictory. How can the utility function be non-satiated if the consumption set is closed, i.e., compact since it is a closed subset in $R^m$. Doesn't compactness and continuity of the utility function guarantee that there exists a bundle within the consumption set that maximizes utility which implies that non-satiation cannot hold?

Answer 1

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.)

Answer 2

The concepts of feasible set and utility should be thought of separately.

Non-satiation just means that there are no "thick" indifference curves - this guarantees that an agent is not indifferent between two close, but different bundles of goods.