Why can't I type a dollar sign in this latex equation?

Question

Consider the following latex document.

\documentclass{article}

\usepackage{tikz}

\begin{document}
Consider the following picture.
\[
  \begin{tikzpicture}
    \node (X) {X}; % this X should be enclosed in a dollar sign to be in math mode
    \node[right of= X] (Y) {Y}; % this Y should be enclosed in a dollar sign to be in math mode
    \draw[->] (X.east) -- (Y.west);
  \end{tikzpicture}
\]
\end{document}

I want to enclose the X and Y in the tikzpicture environment with dollar signs (because I want them typeset in math mode). However, when I attempt to do so, emacs throws an error in the minibuffer reading Math mode started with '\[' cannot be closed with dollar.

What's wrong here?

I'm using GNU Emacs 27.1 with AUCTeX-version 12.3.1.

Note that this is not a latex question since my desired code (below) compiles fine in latex.

\documentclass{article}

\usepackage{tikz}

\begin{document}
Consider the following picture.
\[
  \begin{tikzpicture}
    \node (X) {$X$};
    \node[right of= X] (Y) {$Y$}; 
    \draw[->] (X.east) -- (Y.west);
  \end{tikzpicture}
\]
\end{document}

Answer 1

AucTeX redefines the binding for $ to be the command TeX-insert-dollar. That command checks whether you are in a math environment already and complains if you try to close it the "wrong" way. Doing C-h f TeX-insert-dollar shows you the doc string of the function:

Insert dollar sign.

If current math mode was not entered with a dollar, refuse to
insert one.  Show matching dollar sign if this dollar sign ends
the TeX math mode and ‘blink-matching-paren’ is non-nil.

When outside math mode, the behavior is controlled by the variable
‘TeX-electric-math’.

With raw C-u prefix, insert exactly one dollar
sign.  With optional ARG, insert that many dollar signs.

Note the part that says: If current math mode was not entered with a dollar, refuse to insert one.

BTW, it's not clear to me what the \[ ... \] delimiters are buying you. If you get rid of them, then AucTeX will not complain about the $ signs, and you still have a centered display. CORRECTION: It's not centered but you can do that with \begin{center}...\end{center} and you'll have to add some space before and after it perhaps.

Alternatively, keep the display but change the font of the text of the nodes:

\[
  \begin{tikzpicture}[every text node part/.style={font=\itshape}]
    \node (X) {X};
    \node[right of= X] (Y) {Y};
    \draw[->] (X.east) -- (Y.west);
  \end{tikzpicture}
\]

And yet another way, closer in spirit to the $...$ method but without resorting to $ signs:

\[
  \begin{tikzpicture}
    \node (X) {\ensuremath{X}}; 
    \node[right of= X] (Y) {\ensuremath{Y}}; 
    \draw[->] (X.east) -- (Y.west);
  \end{tikzpicture}
\]