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'inclusion mapping'
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| Title of object: |
inclusion mapping |
| Canonical Name: |
InclusionMapping |
| Type: |
Definition |
| Created on: |
2003-06-26 11:05:02 |
| Modified on: |
2003-06-26 11:07:37 |
| Classification: |
msc:03E20 |
| Synonyms: |
inclusion mapping=inclusion map
inclusion mapping=inclusion |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\renewcommand{\bibname}{References} |
Content:
{\bf Definition} Let $X$ be a subset of $Y$. Then the \emph{inclusion map}
from $X$ to $Y$ is the mapping
\begin{eqnarray*}
\iota: X&\to& Y \\
x&\mapsto& x.
\end{eqnarray*}
In other words, the inclusion map is simply a fancy way to say
that every element in $X$ is also an element in $Y$.
To indicate that a mapping is an inclusion mapping, one usually writes
``$\hookrightarrow$'' instead of ``$\to$'' when defining or mentioning an
inclusion map. In the above definition, we have not used this convention.
However, examples of this convention would be:
\begin{itemize}
\item
Let $\iota:X\hookrightarrow Y$ be the inclusion map from $X$ to $Y$.
\item
We have the inclusion $S^n\hookrightarrow \sR^{n+1}$.
\end{itemize} |
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