Definition:Identification Topology

Definition
Let $\left({S_1, \tau_1}\right)$ 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 $\left({S_1, \tau_1}\right)$ is defined as:
 * $\tau_2 = \left\{{V \in \mathcal P \left({S_2}\right): f^{-1} \left({V}\right) \in \tau_1}\right\}$

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 $\tau_1$.

Also known as
Some sources call $\tau_2$ the quotient topology on $S_2$ with respect to $f$ and $\left({S_1, \tau_1}\right)$'''.

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 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


 * Definition:Final Topology
 * Definition:Quotient Topology


 * Identification Topology equals Quotient Topology on Induced Equivalence