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I note that someone did ask a similar question here (Thin indifference curves), but I didn't fully understand the answer, and my question refers specifically to my textbook.

In my textbook, Jehle & Reny, the axiom of continuity is defined as follows:

Excerpt from Jehle & Reny, page 8

Figure 1.1

Why does continuity alone preclude an indifference region with "thickness"? I would've thought the preference relation for a consumer who is indifferent between all combinations of x1 and x2 still satisfies continuity, and that non-satiation is required to rule out "thick" indifference curves.

Am I misunderstanding something (highly likely)?

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    $\begingroup$ It does not. It just implies that indifference curves are closed sets. $\endgroup$ Jan 5 at 11:41
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    $\begingroup$ Yes, compare the figure with Fig. 1.2., where the boundary lines are no longer dashed, but the curve is still "thick". $\endgroup$
    – VARulle
    Jan 5 at 12:03
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    $\begingroup$ Yes, exactly. Because an indifference curve through a point is the intersection of the corresponding weakly-better-set and weakly-worse-set, and these are closed by the continuity assumption. $\endgroup$ Jan 5 at 13:52
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    $\begingroup$ @MichaelGreinecker Please post answers as answers. $\endgroup$
    – Giskard
    Jan 5 at 15:22
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    $\begingroup$ @Giskard, I will write a polished answer to my own question once I've ironed out my understanding. Good practice for me! $\endgroup$
    – cashman
    Jan 6 at 0:26

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To answer my own question, continuity can allow for thick indifference curves, but it precludes open areas, such as the dotted line shown in the picture.

Continuity guarantees the existence of a closed indifference set, that being the intersection of the weakly-better-than-set and weakly-worse-than-set.

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