topology-Quotient topology,quotient space

Can somebody please explain me the following example given in Munkres, quotient topology?

Let $A={a,b,c}$ and $p$ be map from real line to the set $A$ defined by

$p(x)=a$ if $x>0$, $p(x)=b$ if $x<0$, $p(x)=c$ if $x=0$.

Find the quotient topology on $A$ induced by $p$.

I understand the definition of quotient topology,but couldn't get an idea about how to form the topology given in this example.

How can we relate this example to quotient space?

Here the subsets of $A$ are $\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\{\}$.

by definition of $p$, inverse image of $\{a\}$ is the open interval $(0,\infty)$, inverse image of $\{b\}$ is the open interval $(-\infty,0)$ and inverse image of $\{a,b\}$ is $\Bbb{R}\setminus\{0\}$.

for other subsets we cannot find any open set in $\Bbb{R}$ such that the inverse maps to that open set.

Is this sufficient for us to say that the topology induced by $p$ is a topology on $A$?

You only have to check that $p^{-1}(b)= ]- \infty,0[$ is open, $p^{-1}(a)=[0,+ \infty[$ is open, $p^{-1}(c)= \{0 \}$ is not open, $p^{-1}(\{a,b\})= \mathbb{R}^*$ is open, $p^{-1}(\{a,c\})=[0,+ \infty[$ is not open, etc.

Indeed, $U \subset A$ is open iff $p^{-1}(U)$ is open in $\mathbb{R}$.