Definition:Identification Topology

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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

  • Results about the identification topology can be found here.