Using `\textbf{}` to make also math bold in sentence

Question

Consider the following MWE:

\documentclass{amsart}

\begin{document}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the \textbf{open ball of radius $r$ around $x$} is the set 

\begin{equation}
    B_r(x) := \{y \in M : d(y,x) < r\}.
\end{equation}
\end{document}

Is there any possibility that the math symbols are also bold by default, i.e. without writing explicitly something like \mathbf{a + b}?

Answer 1

Update 2023

With a newer release of LaTeX, there's a much better way to accomplish the task.

\documentclass{amsart}
\AddToHook{cmd/bfseries/after}{\boldmath}
\AddToHook{cmd/normalfont/after}{\unboldmath}
\begin{document}
\section{Bad typesetting}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the 
\textbf{open ball of radius $r$ around $x$} is the set 
\begin{equation}
    B_r(x) := {y \in M : d(y,x) < r}.
\end{equation}
Let's see: \textbf{M \normalfont (M) M}
\section{Good typesetting}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the 
\emph{open ball of radius~$r$ around~$x$} is the set 
\begin{equation}
    B_r(x) := {y \in M : d(y,x) < r}.
\end{equation}
\end{document}

Older answer, don't use unless you have an old TeX distribution.

You might update the definition of \bfseries to also issue \boldmath. However I recommend using \emph for emphasis.

Never leave a blank line before an equation or other math display environment. Use ties, as I show in the second example.

\documentclass{amsart}
\makeatletter
\DeclareRobustCommand\bfseries{%
  \not@math@alphabet\bfseries\mathbf
  \fontseries\bfdefault\selectfont
  \boldmath % <-- added
}
\makeatother
\begin{document}
\section{Bad typesetting}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the 
\textbf{open ball of radius $r$ around $x$} is the set 
\begin{equation}
    B_r(x) := {y \in M : d(y,x) < r}.
\end{equation}
\section{Good typesetting}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the 
\emph{open ball of radius~$r$ around~$x$} is the set 
\begin{equation}
    B_r(x) := {y \in M : d(y,x) < r}.
\end{equation}
\end{document}

Why is it bad using boldface type? Because it's too heavy. Why is it worse to embolden math? Because math symbols have a meaning that also depends on their typographic nature: to a mathematician, a boldface italic ‘r’ is not the same as a medium italic ‘r’.

Answer 2

Update \textbf to also execute \boldmath:

\documentclass{amsart}

\usepackage{mathtools}

\let\oldtextbf\textbf
\renewcommand{\textbf}[1]{\oldtextbf{\boldmath #1}}

\begin{document}
Let $(M,d)$ be a metric space. For any $x \in M$ and $r > 0$, the \textbf{open ball of radius~$r$ around~$x$} is the set 
\begin{equation}
  B_r(x) \vcentcolon= \{y \in M \mid d(y,x) < r\}.
\end{equation}

\end{document}

For some this may change the notation, since there could be a different interpretation of r and r, for example.

---

A wiser choice for the redefinition of \textbf:

\usepackage{letltxmacro}
\LetLtxMacro\oldtextbf\textbf% http://tex.stackexchange.com/q/88001/5764
\DeclareRobustCommand{\textbf}[1]{\oldtextbf{\boldmath #1}}