# inclusion mapping

Definition Let $X$ be a subset of $Y$. Then the from $X$ to $Y$ is the mapping

 $\displaystyle\iota:X$ $\displaystyle\to$ $\displaystyle Y$ $\displaystyle x$ $\displaystyle\mapsto$ $\displaystyle x.$

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 $S^{n}\hookrightarrow\mathbb{R}^{n+1}$.

Title inclusion mapping InclusionMapping 2013-03-22 13:43:08 2013-03-22 13:43:08 Koro (127) Koro (127) 9 Koro (127) Definition msc 03E20 inclusion map inclusion Pullback2