To describe the topology of a quotient space $X/\sim=\bar X$ from the sole definition may be challenging. That's why the we talk about quotient maps, which is a concept that will help us handle quotient topologies more easily
A map $q:X\longrightarrow X’$ is a quotient map when $X’$ has the same topology than when identifying the elements in $X$ that have the same image. This way, if we have a candidate to topology of $X/\sim=\bar X$ (for instance we want to check that $\bar X\simeq X’$), all we need is
- A map $q:X\longrightarrow X’$ that is in accordance with the equivalence set in $X$, that is, $q(x)=q(y)\Longleftrightarrow x\sim y$
- To prove that $q$ is a quotient map
and automatically we'll have that $\bar X\simeq X’$. So our goal right now is to find distinctive properties of quotient maps.