# Why is American and French notation different for open intervals (x, y) vs. ]x, y[?

The Americans and the French use a different notation for open intervals: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?

• The square brackets notation is due to Bourbaki. Commented Oct 30, 2014 at 16:56
• Some discussion on this topic here and here. One commenter suggest that the backward brackets might have been introduced by Bourbaki to prevent confusion with ordered pairs. I am still at a loss for a documented history, but it is at least a part of an old ISO standard. I am not seeing it in the latest standard ISO 80000-2. Commented Oct 30, 2014 at 18:06
• I guess it is intuitive (inclusion/exclusion of the endpoints depends on the direction of the bracket), but I have not found anything written by them stating this or some other motivation explicitly. Commented Oct 30, 2014 at 18:18
• @J.W.Perry Thanks! I had indeed went through the standard for that reason, see notation 2-6.10 last column. Commented Oct 30, 2014 at 23:54
• The ] [ notation is used in Belgium too, also in the Dutch-speaking part. Commented Feb 27, 2023 at 13:27

Notation $()$ is traditional, and $].[$ was introduced by Bourbaki.

Much of the Bourbaki notations and terminology became standard, but English speaking people are the most conservative ones in this respect:-) (Recall the history of the metric system:-)

Another example of the same is "injection", "surjection", "bijection". Many English authors still write "one-to-one", "onto" and "one-to-one and onto".

Another example: Bourbaki taught us that "positive" is $\geq 0$, and "strictly positive" is $>0$.

But many people still prefer "positive" to mean $>0$ and "non-negative" for $\geq0$.

Remark. I am educated in Ukraine in 1970-s, and I experienced a strong influence of Bourbaki on education. But I still like $(,)$, perhaps just for aesthetic reasons.