This question explains how to redefine the \emph
command so that the text is bold. When mathematical text is included with emphasized text, I have been using the following solution:
\documentclass{report}
\usepackage{bm}
\DeclareTextFontCommand{\emph}{\bfseries}
\begin{document}
\noindent Here the mathematical text is not bold: \\
\noindent An \emph{$n$-dimensional chart} of a topological space $M$ is a pair
$(U,\mathbf{x})$, where $U$ is an open subset of $M$ and $\mathbf{x}: M \supset U
\to \mathbf{x}(U)\subset \mathbf{R}^n$ is a homeomorphism. \\
\noindent Here is the corrected version: \\
\noindent An \emph{$\bm{n}$-dimensional chart} of a topological space $M$ is a pair
$(U,\mathbf{x})$, where $U$ is an open subset of $M$ and $\mathbf{x}: M \supset U \to
\mathbf{x}(U)\subset \mathbf{R}^n$ is a homeomorphism.
\end{document}
My question is: how can you further redefine \emph
so that mathematical text is automatically bold, using \bm
?