inclusion mapping

Definition Let $X$ be a subset of $Y$ . Then the 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:

• Let $\iota:X\hookrightarrow Y$ be the inclusion map from $X$ to $Y$ .
• We have the inclusion $S^n\hookrightarrow \sR^{n+1}$ .