Making space after comma larger than the spacing around a binary relation sign

Question

Consider the following code:

\RequirePackage{fix-cm}
\documentclass[10pt,conference,compsoc,letterpaper]{IEEEtran}
\usepackage{amsfonts}
\usepackage{amsmath}
\newcommand{\Foo}{\mathsf{Foo}}
\newcommand{\Blob}{\mathsf{Blob}}
\newcommand{\powerset}{\mathfrak{P}}
\begin{document}
Let \(A\) \(=\) \(\Bigl\{g\colon \bigl(\Foo\mkern-1mu\times\mkern-1mu\powerset((\Blob{\times}\Foo)^2)\bigr)\to\Foo\to\mathbb{N}_{\ge 0}\,\allowbreak\Bigm|\,\allowbreak\forall\,a{\in}\Foo,\,{\leadsto}\mkern-1mu\subseteq\mkern-1mu(\Blob{\times}\Foo)^2\colon\mkern1mu \sum_{b\in\Foo}g(a,{\leadsto})(b)=\bigl\lvert\{t{<}n\mid l_t\mkern-1mu=\mkern-1mua \land {\leadsto}\mkern-1mu=\mkern-1mu{\rightarrow_t}\}\bigr\rvert\Bigr\}\).
\end{document}

The output is:

It is ugly in the second line: the spacing around ⊆ is larger than the space after a∈Foo,. It should be the other way round: the space before ↝ should be larger than after it. Reason: the all-quantifier ∀ logically binds two variables: a and ↝, where a ranges over Foo, and ↝ ranges over subsets of (Blob × Foo)². Moreover, the spacing around ⊆ must remain larger than the spacing around × in (Blob×Foo)².

Of course, I could adjust the spacing manually via \mkern, \,, \, \hspace, \!, etc. in an ad-hoc way, but I've got tons of similar formulas. Therefore, I need a more general solution. It should also work when the column width is changed for the whole document. Ideas?

Answer 1

I find the comma is the least of your problems. If you want two variables being quantified, use two quantifiers.

You're trying to override the standard math spacing in all possible ways, mostly wrong. The result is something where the spaces seem placed at random: the reason is that space around relation and operation symbols is flexible, whereas the spaces after punctuation isn't. However you need such flexibility in order to have justification: an unsolvable problem.

It's even “more unsolvable” if you want a solution that work also when the typesetting is in a single column. Mathematics in two column format needs many compromises: it only works decently well with short formulas.

Moreover, your readers will not be able to interpret what you're trying to tell them. Long set descriptions should better be in words: your text, even if spaced correctly, is very difficult to parse.

I'm not sure what sort of math object is g, so I call it an extrablob.

\documentclass[10pt,conference,compsoc,letterpaper]{IEEEtran}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{mathptmx} % IEEEtran does \usepackage{times}

\usepackage{lipsum} % just for the example

\newcommand{\Foo}{\mathsf{Foo}}
\newcommand{\Blob}{\mathsf{Blob}}
\newcommand{\powerset}{\mathfrak{P}}
\newcommand{\varlto}{{\leadsto}} % \leadsto as a variable
\newcommand{\varto}{{\rightarrow}} % \rightarrow as a variable

\begin{document}

\lipsum[3]

Let \(A\) be the set of extrablobs
\[
g\colon \Foo\times\powerset((\Blob\times\Foo)^2)\to\Foo\to\mathbb{N}_{\ge 0}
\]
such that, for all \(a\in\Foo\) and all
\(\varlto\subseteq(\Blob\times\Foo)^2\),
\[
\sum_{b\in\Foo}g(a,\varlto)(b)=
\bigl\lvert\{t<n \mid l_t=a \land \varlto=\varto_t\}\bigr\rvert.
\]
\lipsum[4]

\end{document}