inclusion mapping

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

$\iota :X$$\to$$Y$

$x$$\mapsto$$x.$

In other words, the inclusion map is simply a fancy way to saythat 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 aninclusion map. This hooked arrow symbol $\hookrightarrow$ can beseen 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 $S^n\hookrightarrow {ℝ}^{n+1}$.