# 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

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Functions - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.8$: Quotient spaces: Definition $3.8.1$