In the following question here the notation $c\in ]a,b[$ is used. What does this mean? I have never seen it before.

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The notation is especially common when there is the possibility of confusing the ordered pair $(a,b)$ with the interval $(a,b)$. Using $]a,b[$ for the interval eliminates this possibility. –  Jason DeVito Nov 26 '12 at 0:47
To complement the answers below, square brackets are used to indicate ends of intervals, so $[a,b]$, $[a,b[$, $]a,b]$ and $]a,b[$ mean the closed interval, the half-closed (or "half-open") interval with $b$ excluded, the half-closed interval with $a$ excluded, and the open interval, respectively, all with end-points $a$ and $b$. As already mentioned, this notation (due to Bourbaki) avoids ambiguity by being different from the notation used for ordered pairs. –  Andres Caicedo Nov 26 '12 at 0:51

$]a,b[$ is somtimes used to denote the open interval $(a,b)$ i.e. $$]a,b[ = \{x \in \mathbb{R}: x > a \,\,\,\& \,\,\, x <b\}$$

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$\;\;]\,a,b\,[\;\;$ is used by some to denote the open interval $(a,b)$.

Put differently, $$\,]\,a,b\,[\,\, = \{x \in \mathbb{R}:\; a\,<x\,<b\},$$ which reads, "the set of all real numbers greater than $a$ but less than $b$".

This notation helps to distinguish the open interval $(a, b)$ from the ordered pair $(a, b)$.

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I LOVE () INSTEAD OF ][ –  B. S. Aug 27 '13 at 5:58
The expression $c\in ]a,b[$ means "$c$ belongs to the open interval from $a$ to $b$". Another (more common) way to denote the same thing is $c \in (a,b)$.