inclusion mappingInclusionMapping2003-06-26 11:05:022004-01-31 04:24:06Definitionmapping% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition} Let $X$ be a subset of $Y$. Then the {\bf 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. This hooked arrow symbol $\hookrightarrow$ can be
seen as combination of the symbols $\subset$ and $\to$.
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}