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What is a Jordan Region? I can't find the definition anywhere. The question asked, if $E\subset \mathbb{R}^n$, bounded and with finitely many accumulation points, then $E$ is Jordan region.

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  • $\begingroup$ I think this refers to Jordan measurable sets. $\endgroup$
    – user147263
    Apr 25, 2015 at 18:36

1 Answer 1

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From pg. 384 of An Introduction to Analysis (3rd Edition) by William R. Wade:

12.3 Definition. Let $E$ be a subset of $\mathbb{R}^n$. Then $E$ is said to be a Jordan region if and only if given $\epsilon > 0$ there is rectangle $R \supseteqq E$, and a grid $\mathcal{G} = \lbrace R_{1}, \ldots, R_{p} \rbrace$ on $R$, such that

$$ V(\partial E; \mathcal{G}) := \sum_{R_{j} \bigcap \partial E \neq \emptyset} |R_{j}| < \epsilon. $$

... Thus a set is a Jordan region if and only if its boundary is so thin that it can be covered by rectangles whose total volume is as small as one wishes.

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  • $\begingroup$ Can you give an example in R^2 or R^3? $\endgroup$ Jan 3, 2021 at 13:42

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