Definition:Inclusion Mapping
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Definition
The inclusion mapping $i_S: S \to T$ is a mapping on a set $S$ defined when $S \subseteq T$:
- $i_S: S \to T: \forall x \in S: i_S \left({x}\right) = x$
It can be seen that the inclusion mapping $i_S$ is the restriction to $S$ of the identity mapping $I_T: T \to T$.
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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 $T$.
However, beware of confusing this with the use of the term canonical injection in the field of abstract algebra.
Notation
Beware the notation. Always be sure you understand what is being used.
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:
- $\iota \left({n}\right) = \begin{cases} 1 & : n = 1 \\ 0 & : n \ne 1 \end{cases}$
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$: Example $5.4$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
