# Definition:Identification Topology

## Definition

Let $\struct {S_1, \tau_1}$ be a topological space.

Let $S_2$ be a set.

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

The **identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$** is defined as:

- $\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$

That is, it is the set of all subsets of $S_2$ whose preimages under $f$ are elements of $\tau_1$.

The **identification topology** is seen to depend both on $f$ and $\tau_1$.

### Identification Mapping

The mapping $f: S_1 \to S_2$ in this context is called the **identification mapping**.

## Also known as

Some sources call $\tau_2$ the **quotient topology** on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$**.**

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 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.

## Also see

- Identification Topology is Topology
- Identification Mapping is Continuous
- Existence and Uniqueness of Identification Topology
- Identification Topology is Finest Topology for Mapping to be Continuous

- Results about
**the identification topology**can be found here.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces: Definition $3.8.1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions