Factoring Mapping into Quotient and Injection

Theorem

Any mapping $f: S \to T$ can be uniquely factored into a quotient mapping, followed by an injection.

Thus:

$f = h \circ q_{\mathcal R_f}$

where:

• $q_{\mathcal R_f}: S \to S / \mathcal R_f : q_{\mathcal R_f} \left({s}\right) = \left[\!\left[{s}\right]\!\right]_{\mathcal R_f}$
• $h: S / \mathcal R_f \to T : h \left({\left[\!\left[{s}\right]\!\right]_{\mathcal R_f}}\right) = f \left({s}\right)$

This can be illustrated using a commutative diagram as follows:

Proof

The mapping $q_{\mathcal R_f}: S \to S / \mathcal R_f$ follows from the definition of quotient mapping.

The mapping $h$ is justified by Existence of Renaming Mapping.

$\blacksquare$

Also see

• Factoring Mapping into Surjection and Inclusion

• Quotient Theorem for Surjections
• Quotient Theorem for Sets

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Theorem $10$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $4.10 \ \text{(ii)}$