Strange interval notation:$\;]a,b[$

Question

In one older book I found this notation for an interval:

$$]a,b[$$

What does it mean?

Answer 1

It is an alternative characterisation of the open interval $(a,b)$. So it holds that $x \in (a,b) \iff x \in ]a,b[$

Wikipedia generalizes the above with equivalent notions for closed and half open intervals with the following image:

This can be found under the “Notations for intervals” subheading on the Wikipedia page for intervals.

This alternative notation was introduced by the French mathematician Nicolas Bourbaki to avoid ambiguity between the interval $(a,b)$ and the ordered pair $(a,b)$.

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Note that there is also an alternative use of this notation by a minority of authors, so do be careful. The above article states the following:

“Some authors such as Yves Tillé use $]a, b[$ to denote the complement of the interval $(a,b)$; namely, the set of all real numbers that are either less than or equal to $a$, or greater than or equal to $b$.”

Whilst this isn’t the normal use of this notation, it’s worth keeping in mind that this is also sometimes used. If it isn’t explained in the textbook, then it may be something you need to infer from the context.