Factoring Mapping into Surjection and Inclusion

Theorem

Every mapping $f:S \to T$ can be uniquely factored into a surjection $g$ followed by the inclusion mapping $i_T$.

That is, $f = i_T \circ g$ where:

$g: S \to \Img f: \map g x = \map f x$

$i_T: \Img f \to T: \map {i_T} x = x$

This can be illustrated using a commutative diagram as follows:

$\quad\quad \begin{xy}\xymatrix@L+2mu@+1em { S \ar@{-->}[r]^*{g} \ar[rd]_*{f = i_T \circ g} & \Img f \ar@{-->}[d]^*{i_T} \\ & T }\end{xy}$

Proof

From Surjection by Restriction of Codomain, $f: S \to \Img f$ is a surjection.

The mapping $g: S \to \Img f$ where $\map g x = \map f x$ is therefore also clearly a surjection.

The mapping $g: S \to \Img f: \map g x = \map f x$ is clearly unique, by Equality of Mappings.

From Inclusion Mapping is Injection, $i_T: \Img f \to T$ is an injection.

Likewise, the mapping $i_T: \Img f \to T : \map {i_T} x = x$ is also unique, by its own definition.

$\blacksquare$

Also see

• Factoring Mapping into Quotient and Injection

• Quotient Theorem for Surjections
• Quotient Theorem for Sets

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Factoring Functions