# Real line with positive infinity adjoined: typeset more compactly

For a document I am working on I frequently need to refer to the real line with positive infinity adjoined (but not negative infinity). I have been typesetting this as follows

\mathbb{R} \cup \{+\infty\}

This takes up a lot of horizontal space. Is there a way to get a smaller plus symbol? Or to cramp the symbols together a bit more? I just feel a bit bothered that, in the expressions where this space occurs, it doesn't really register visually as a discrete entity unto itself.

• you could use {\cup} to lose the binop spacing but can't you define this to be R_\infinity or some such, and then just use the defined symbol? – David Carlisle Oct 17 '20 at 14:48
• Welcome to tex.sx. I'm not competent to say what this change might imply to the meaning, but to set the \cup as an "ordinary " character, eliminating all space, wrap it in braces: {\cup}. – barbara beeton Oct 17 '20 at 14:48
• Honestly it doesn't bother me. I'd use larger parentheses \bigl(...\bigr). You can lose the spacing around \cup by enclosing it in braces {\cup}. – campa Oct 17 '20 at 14:49
• @DavidCarlisle: Thanks! I thought about using something like R_infinity, but felt it should really be R_{+infinity} to avoid confusion with the one point compactification, and the latter looked too clunky for me. Anyway your other suggestions here and in your answer below are very helpful, so thanks! – Michael Oct 17 '20 at 14:58

Personally I think I'd use the form you show, but if using it a lot make it a defined term such as R' then just use that defined notation.

But if you want to squeeze it, perhaps

\documentclass{article}

\usepackage{amsmath,amsfonts,amssymb}

\begin{document}

$C_0(\mathbb{R}{\scriptstyle \cup \{+ \infty\}}) \rtimes \mathbb{R}$
\end{document}

• Neat, I may use something like this! Thanks for indulging this rather frivolous question! :) – Michael Oct 17 '20 at 14:59

I'd go with a special symbol that conveys the idea:

\documentclass[twocolumn]{article}
\usepackage{amsmath,amssymb}

\DeclareRobustCommand{\upcomp}{{{-}\!\!{\bullet}}}
\DeclareRobustCommand{\downcomp}{{{\bullet}\!\!{-}}}

\newcommand{\RR}{\mathbb{R}}
\newcommand{\RRup}{\RR_{\upcomp}}
\newcommand{\RRdown}{\RR_{\downcomp}}

\begin{document}

We define $\RRup=\RR\cup\{+\infty\}$ and $\RRdown=\RR\cup\{-\infty\}$
with the topology induced by the usual extended real line.

Now we can use $C_0(\RRup)$ that takes less space.

\end{document}


Alternative with \mapsfromchar of stmaryrd:

\documentclass[twocolumn]{article}
\usepackage{amsmath,amssymb,stmaryrd}

\DeclareRobustCommand{\upcomp}{{-\mapsfromchar}}
\DeclareRobustCommand{\downcomp}{{\mapstochar-}}

\newcommand{\RR}{\mathbb{R}}
\newcommand{\RRup}{\RR_{\upcomp}}
\newcommand{\RRdown}{\RR_{\downcomp}}

\begin{document}

We define $\RRup=\RR\cup\{+\infty\}$ and $\RRdown=\RR\cup\{-\infty\}$
with the topology induced by the usual extended real line.

Now we can use $C_0(\RRup)$ that takes less space.

\end{document}


• Well worth considering. Thanks! – Michael Oct 17 '20 at 15:23