# Definition:Inclusion Mapping

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## Definition

Let $T$ be a set.

Let $S\subseteq T$ be a subset.

The **inclusion mapping** $i_S: S \to T$ is the mapping defined as:

- $i_S: S \to T: \forall x \in S: \map {i_S} x = x$

## Also known as

This is also known as:

- the
**canonical inclusion of $S$ in $T$** - the
**(canonical) injection of $S$ into $T$** - the
**embedding of $S$ into $T$** - the
**insertion of $S$ into $T$**.

However, beware of confusing this with the use of the term canonical injection in the field of abstract algebra.

## Notation

Some authors use $i_S$ (or similar) for the identity mapping, and so use something else, probably $\iota_S$ (Greek *iota*), for the inclusion mapping.

Another notation is:

- $f: S \subseteq T$

or

- $f: S \stackrel f {\subseteq} T$

The symbol $\iota$ is also used in the context of analytic number theory to denote the Identity Arithmetic Function:

- $\map \iota n = \begin{cases} 1 & : n = 1 \\ 0 & : n \ne 1 \end{cases}$

Some sources use the same symbol for the **identity mapping** as for the inclusion mapping without confusion, on the grounds that the domain and codomain of the latter are different.

## Also see

- Results about
**inclusion mappings**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Composition of Functions - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.14$: Composition of Functions: Theorem $14.6$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Some special types of function - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Example $5.4$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings