inclusion mapping

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

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

$x$ $\mapsto$ $x.$

In other words, the inclusion map is simply a fancy way to 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 $S^n\hookrightarrow {ℝ}^{n+1}$.

Title	inclusion mapping
Canonical name	InclusionMapping
Date of creation	2013-03-22 13:43:08
Last modified on	2013-03-22 13:43:08
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	9
Author	Koro (127)
Entry type	Definition
Classification	msc 03E20
Synonym	inclusion map
Synonym	inclusion
Related topic	Pullback2