How to properly typeset math intervals

In the TeXbook, Knuth refers to people using “]a,b[” notation for open intervals as “perverse mathematicians” (page 171, exercise 18.14). I don't fully agree with the adjective, but I find the notation very awkward nonetheless.

There is already a package for this, which avoids reinventing the wheel:

\documentclass{amsart}
\usepackage{interval}

\begin{document}

\begin{equation}
\interval[open]{-\pi}{0}
 \qquad
{-\pi} % this needs braces because it is between Ord atoms
 \qquad
\int_{-\pi}^0
 \qquad
\interval{-\pi}{0}
\end{equation}

\begin{equation}
x\in\interval[open]{-\pi}{0}
\end{equation}

\end{document}

The middle -\pi needs braces, but it's a very different problem.

See the documentation for interval to learn about other options.

---

Note about Werner's solution

Consider the following code and compile it to see what results:

$x \in ]{-\pi},0[$

$x \in \mathopen]-\pi,0\mathclose[$

(properly embed it in a standard document).

It should be clear that simply bracing -\pi is not sufficient.

TeX will provide the correct spacing if you inform it that you are using ] and [ in a non-standard way, which can be done by means of \mathopen and \mathclose:

\[ \mathopen]-\pi,0\mathclose[ \]

This tells TeX exactly what is going on.

More precisely, TeX assumes that [ is an Open[ing] atom and the ] is a Close[ing] one (and I sympathize with it! ;-) If you write

\[ ]-\pi,0[ \]

TeX will build the following list of atoms: Close, Bin, Ord, Punct, Ord, Open; this leads it to compute the difference between ] and \pi, as Werner has already said; indeed, compare the above with the formula

(a+b)-c

where the minus is the second Bin atom in the resulting sequence Open, Ord, Bin, Ord, Close, Bin, Ord.

Note that Werner’s suggestion, that is,

]{-\pi},0[

yields Close, Ord{…}, Punct, Ord, Open, which does not correspond to the intended meaning, although it gives the correct spacing (but only, of course, in this particular case—see @egreg’s answer), as you can check in the table on p. 170 of The TeXbook.

IMHO, however, the best thing of all is to define an abstract command: it’s very easy to do so using the mathtools package and its \DeclarePairedDelimiterX command; the command defined in this way provides easy means to deal with size issues: see the documentation of the mathtools package, subsection 3.6, for details.

Here is a complete example that illustrates both solutions:

% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly 
                                 % declare the paper format.

\usepackage[T1]{fontenc}         % Not always necessary, but recommended.
% End of standard header.  What follows pertains to the problem at hand.

\usepackage{mathtools}

% [ bracket matching
\DeclarePairedDelimiterX{\openinterval}[2]{]}{[}{#1,#2}
% ] bracket matching



\begin{document}

Wrong:
\[ ]-\pi,0[ \]
Direct method, not recommended:
\[ \mathopen]-\pi,0\mathclose[ \]
With an abstract command:
\[ \openinterval{-\pi}{0} \]

Variants of the abstract command:
\begin{align*}
    &\openinterval*{-\frac{\pi}{2}}{+\frac{\pi}{2}}
            && \text{auto-resizing;}  \\
    &\openinterval[\bigg]{-\frac{\pi}{2}}{+\frac{\pi}{2}}
            && \text{with optional argument for size specification.}
\end{align*}

\end{document}

And here is the output it produces:

TeX considers your usage as ] minus \pi. Remove this ambiguity by using (say) {-\pi}:

\documentclass{article}

\begin{document}

\[
  ]{-\pi},0[ \qquad -\pi \qquad \int_{-\pi}^0
\]

\end{document}

For more complex interactions, consider using Gustavo's answer. For example, to show membership or another relation/operator with respect to the set, you'll have to use an empty atom for proper spacing:

x \in{} ]{-\pi},0[ {}\ni x