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I noticed that in many documents you encounter $f: X \to Y$ instead of $f \colon X \to Y$. Another example are quantified expressions: $\forall x: P (x)$ versus $\forall x \colon P(x)$. I find the \colon visually more pleasing, however I sometimes think I seem to be the only one, given the overwhelming amount of documents that seem to use :. Is there a rule of thumb to decide where you should use either \colon or :?

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Both : and \colon typeset a colon, but \colon is a punctuation symbol, while : is considered as a relation symbol as regards to spacing.

The main use of : is in set descriptions

\{\, x : x \notin x \,\}

(somebody uses \mid for this, where a simple | would be wrong; thin spaces after \{ and before \} are recommended by Knuth, be consistent in using them or not).

Conversely, \colon should be used for mappings

f\colon A \to B

but unfortunately many writers don't make this distinction and use :, getting a wrong spacing.

The rule to follow is just the same: use \colon when it's a "punctuation colon", use : when it represents a relation between what's at its left and at its right. In something like "for all x:" I would consider the colon as punctuation, so \forall x\colon

Note that the amsmath package changes the definition of \colon so that it's not exactly the same as a punctuation symbol:

\renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}

This adds a bit of space before the colon, which seems more right than with a simple \mathpunct.

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    And use : for ratios. e.g. $x:y:z = 3:4:5$. – Leo Liu Dec 10 '11 at 11:25
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    @LeoLiu Alternatively, \mathbin{:} could be used. Which one is a matter of personal taste. Most important is consistency across the document. – egreg Dec 10 '11 at 11:32
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    How does one distinguish between "punctuation" and "relation"? The notation f\colon A\to B does denote a relation between f and A\to B, namely the relation "has type". And I don't see in what sense the colon in \{\, x : x\notin x \,\} denotes a "relation" between x and x\notin x. – Mike Shulman Apr 11 '12 at 10:29
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    @MikeShulman It's a matter of conventions; usually the colon for maps is considered as punctuation, the one in set descriptions as a relation symbol. Be coherent in your document. – egreg Apr 11 '12 at 12:30
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    Just for clarity, the usual recommendation to use \, after \{ and before the matching \} is only in case the set involves a set descriptor (with the colon). By contrast, use \{a, b, c\} without the thin spaces in case of an explicit enumeration of elements. – murray Jun 27 '12 at 21:00
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If you're trying to typeset a variable-has-type colon in type theory, you want {:} or \mathord{:} (they display the same). For example, you'll get a nice looking STLC identity function with $\lambda x {:} A . x$.

See this page for a more general discussion.

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