Definition:Quotient Mapping (Topology)

Definition

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a surjective continuous mapping.

Let $f$ fulfil this condition:

For all $U \subseteq S_2$, if $f^{-1} \sqbrk U$ is open in $T_1$, then $U$ is open in $T_2$.

Then $f$ is a quotient mapping.

Also known as

Some texts prefer the brief form quotient map.

Also see

• Definition:Quotient Topology

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $2$: Topological Spaces and Continuous Functions: $\S 22$: The Quotient Topology