Definition:Identification Topology

Definition

Let $\left({X, \vartheta_1}\right)$ be a topological space.

Let $Y$ be a set.

Let $f: X \to Y$ be a mapping.

Then there exists a topology $\vartheta_2$ on $Y$ such that:

• $f: X \to Y$ be a continuous mapping;
• $\vartheta_2$is the finest topology on $Y$ for which $f: X \to Y$ is continuous.

That is:

$\vartheta_2 = \left\{{V \in \mathcal P \left({Y}\right): f^{-1} \left({V}\right) \in \vartheta_1}\right\}$

This topology $\vartheta_2$ is the identification topology on $Y$ with respect to $f$ and $\left({X, \vartheta_1}\right)$.

It is seen to depend both on $f$ and $\vartheta_1$.

Its existence and uniqueness is clear by recognition of the fact that for any $V \subseteq Y$, either $f^{-1} \left({V}\right) \in \vartheta_1$ or $f^{-1} \left({V}\right) \notin \vartheta_1$.

Identification Map

The mapping $f: X \to Y$ in this context is called the identification map.

Also see

• Quotient Topology.

Some sources call $\vartheta_2$ the quotient topology on $Y$ with respect to $f$ and $\left({X, \vartheta_1}\right)$.

This is reasonable, as we can construct the induced equivalence from any mapping $f$ and thence consider the identification topology as the quotient topology.

From Identification Topology Equals the Quotient Topology on Induced Equivalence, it can be seen that they are in fact one and the same thing, but seen from a different angle.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions