Definition:Quotient Topology

Let $$T = \left({A, \vartheta}\right)$$ be a topological space.

Let $$A'$$ be a set.

Let $$f: A \to A'$$ be a surjection.

Then the quotient topology on $$A'$$ by $$f$$ is:


 * $$\vartheta' = \left\{{U \subseteq A': f^{-1} \left({U}\right) \in \vartheta}\right\}$$.

It is alternatively called the identification topology on $$A'$$ by $$f$$.

Quotient Space
The Quotient Topology is a Topology.

Thus it follows that $$T' = \left({A', \vartheta'}\right)$$ is a topological space.

It can be called the quotient space of $$A$$ by $$f$$.

Identification Map
The mapping $$f: A \to A'$$ in this context is called the identification map.

It is immediately seen to be continuous by the definition of $$\vartheta'$$.