Define pretty sets in LaTeX; esp., how to do the condition-separator?

Question

E.g., I have this set:

\{ \alpha \in \Sigma^\omega | \exists^\omega n : \alpha [0,n] \notin \Sigma^* a_1 \Sigma^* a_2 \dotsb a_n \Sigma^* \}

Here, I use | as the separator for the condition part. But LaTeX doesn't add any spaces around it, thus it looks a bit strange/wrong. I could manually add some space here but I wondered if there is some more "correct" way. E.g. also to make it behave similar as : (which I actually want to behave different here; in this case, : adds too much space in front of it for my taste).

Answer 1

I sometimes do something a little more evil

\documentclass[a4paper]{memoir}
\usepackage{mathtools}
\DeclarePairedDelimiterX\Set[2]{\lbrace}{\rbrace}%
 { #1 \,\delimsize|\, #2 }
\begin{document}
\[
\Set*{ x }{ x>0 }
\]
\end{document}

and thus hide the symbol inside the construction. Both a starred and non-starred \Set is created with \Set* autoscaling braces and vertical line is used.

Answer 2

The braket package contains a \set command which does what you want :

\set{\alpha \in \Sigma^\omega | \exists^\omega n : \alpha [0,n] \notin \Sigma^* a_1 \Sigma^* a_2 \dotsb a_n \Sigma^*}

It also defines \Set, where the first | is expandable, which allows to properly typeset sets like

\Set{x| x\in\mathbb R, |x|<\frac12 }

Answer 3

You should use \mid instead of |. And use \colon instead of : after the existential quantifier (although I would just put a thinspace).