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Infinity is defined as something that is uncountable. But every where I see the symbol of Infinity as a closed loop. My professor said, that it symbolizes that the boundary of Infinity is determined but it isn't so. I too agree with him. The Infinity symbol must be open curve rather than closed, something like this:
enter image description here
I would like to define and use this symbol for Infinity. How do I achieve it?

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5  
I can't agree with your professor's opinion. This symbol doesn't denote infinity when referred to sets, but just a mathematical concept used in analysis. –  egreg Jul 23 at 13:59
2  
If you want to be understood by other mathematicians, use established symbols. Or do you also write ":" instead of "2" because the colon better represents two-hood? --That beng said, you might add a small white blob on top of the standard $\infty$ symbol (assuming you only write on white background) –  Hagen von Eitzen Jul 23 at 14:22
1  
Adding infinity gives a one-point compactification of the real numbers, which the closed shape nicely suggests. –  Mass Jul 23 at 15:46
    
I agree with @egreg (and not with your professor). It is just a symbol used traditionally. –  Harish Kumar Jul 25 at 2:28

3 Answers 3

Both symbols are available in Unicode:

  • U+221D: ∝ PROPORTIONAL TO
  • U+221E: ∞ INFINITY

It is not a good idea to replace the well-established symbol for infinity, which even many non-mathematicians will understand, by a different symbol, a relational operator, which is used for a complete different purpose. This is called obfuscating and will irritate the user at best.

Of course LaTeX can print both symbols:

\documentclass{article}
\usepackage{amsfonts}% or latexsym, amssymb, mathabx, txfonts, pxfonts, wasysym
\begin{document}
\[ \infty \propto \]
\end{document}

Result

If \propto is used as symbol for infinity, then it needs a set of curly braces to get rid of the \mathrel property, e.g.:

\newcommand*{\myinfinity}{{\propto}}

Example for LuaTeX/XeTeX with Unicode math fonts. The example also shows different input methods for the symbols:

\documentclass{article}
\usepackage{unicode-math}
\begin{document}
\noindent
% Latin modern
\[ \infty \propto \]
\setmathfont{Asana-Math.otf}
\[ ∞ ∝ \]
\setmathfont{xits-math.otf}
\[ ∞ ∝ \]
\setmathfont{texgyrepagella-math.otf}
\[ ^^^^221e ^^^^221d \]
\end{document}

Result LuaTeX/XeTeX

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\documentclass{article}
\usepackage{color}
\newcommand\myinfty{{\infty%
\mathchoice%
{\llap{\color{white}\vrule width 0.2em height 0.3em depth -0.15em}}%
{\llap{\color{white}\vrule width 0.2em height 0.3em depth -0.15em}}%
{\llap{\color{white}\vrule width 0.2em height 0.22em depth -0.1em}}%
{\llap{\color{white}\vrule width 0.2em height 0.15em depth -0.09em}}%
}}
\begin{document}  
In Text: $\sum_{n=1}^\myinfty \frac1n = \myinfty$
and $\myinfty_{\myinfty_\myinfty}^{\myinfty^\myinfty} $.
In display:
\[\sum_{n=1}^\myinfty \frac1n = \myinfty\]
\[\myinfty_{\myinfty_\myinfty}^{\myinfty^\myinfty}\]
\end{document}
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I took the liberty of removing the pdftex option to color, which shouldn't be given; also I changed $$...$$ into the more correct \[...\]. And +1. –  egreg Jul 23 at 14:56

You can create your own copy of used math font and you can do changes here by fontforge program, for example.

Or you can redefine the \infty macro, for example by such way:

\let\oriinfty=\infty
\def\infty{{\oriinfty\llap{\vrule height.24em depth-.2em width.2em}}}

This example does the black rule over the infinity char in the appropriate place. Now, it is your exercise to do this rule not black but white. You can use color or xcolor package and set the color statement for the \llap{...} part of my macro.

It is more complicated to do this working in scriptstyle and scripscriptstyle too. Read the documentation about \mathchoice TeX primitive for this.

Note: I don't evaluate here if this is good or bad decision to change the infinity character.

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