Peano's dot notation

Question

I am wondering what is the best way to typeset Peano's dot notation.

It is a subset symbol with dots on both sides. For example, for 4/3 I write: ::\supset :\!.. However the spacing and the size of the dots are not correct.

Answer 1

The period is slightly bigger than the dots in the colon (at least in the Computer Modern fonts, other fonts may differ).

You should also center the colon with respect to the math axis.

\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{graphicx}

% Peano colon
\newcommand{\pc}{{\mspace{-1mu}\mathop:\mspace{-1mu}}}
% Peano period
\newcommand{\pp}{{%
  \mspace{-1mu}%
  \sbox0{$:$}\sbox2{$\mathop:$}%
  \raisebox{\dimexpr\ht2-\ht0}{\scalebox{0.99}{$.$}}%
  \mspace{-1mu}%
}}
% Peano implies
\newcommand{\pim}{{\mspace{-1mu}\bm{\supset}\mspace{-1mu}}}

\begin{document}

$d(x,y) \mathrel{\pc\pc\pim\pc\pp} D$

\end{document}

Answer 2

Improving egreg's answer in order to remove arbitrariness and to simplify the code, I finally get:

\documentclass[preview]{standalone}

\usepackage{amsmath, adjustbox}

% Defines specific colon, period and implies
\newcommand{\pc}{{:}}
\newcommand{\pp}{\adjustbox{clip,trim=0pt 0pt 0pt {.5\totalheight}}{$\pc$}}
\newcommand{\pim}{{\boldsymbol{\supset}}}

\begin{document}
    \[P \mathrel{\pp\pim\pc\pp} Q\]
\end{document}

Answer 3

Here there is my humble solution. The output, for my opinion, seem very nice the distance of the dots (they are the same), as from this example, using smallmatrix with amsmath package. The dots are also adapted to the supset symbol.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}
$d(x,a)<r\begin{smallmatrix} .&.\\ .&. \end{smallmatrix}\mkern-7mu\supset \mkern-7mu\begin{smallmatrix} .&\\ .&. \end{smallmatrix}d(x,a)>r$
\end{document}