What is `\mathopen{}` for?

Question

On mathtools manual, page 27, there is an example defining a command \Set:

% just to make sure it exists
\providecommand\given{}
% can be useful to refer to this outside \Set
\newcommand\SetSymbol[1][]{%
    \nonscript\:#1\vert
    \allowbreak
    \nonscript\:
    \mathopen{}}
\DeclarePairedDelimiterX\Set[1]\{\}{%
    \renewcommand\given{\SetSymbol[\delimsize]}
    #1
}

What is this \mathopen{} for? I read the answer of this question, which involves \mathopen along with some explanations, but still don't understand it here.

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Edit:

I just saw that on page 26 in a previous example there is some explanation:

Does this mean that \mathopen{} here is basically the same as \mathopen\delimsize\vert (I'm not sure if this is a valid combination)?

Answer 1

Some people write ]-1,1[ to denote the open interval from -1 to 1. The default TeX math status of the characters [ and ] is mathopen and mathclose, respectively. To properly typeset the open interval mentioned in the first sentence, it's a really good idea to input it as

\mathopen{]} -1 , 1 \mathclose{[}

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Observe that if one fails to inform TeX that the ] character should be treated as having type mathopen, TeX is led to interpret the - ("minus") character as a binary instead of as a unary operator.

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The term \mathopen{} inserts an empty atom of type mathopen into the input stream. This is helpful in case the material that comes after \vert starts with a unary operator such as +, -, \pm, etc. (The default math status of \vert is mathord.)

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\documentclass{article}
\begin{document}
$]-1,1[$ \qquad $\mathopen{]}-1,1\mathclose{[}$
\end{document}