Consider the following pushout diagram in the category $\text{Top}$:
$$ \require{AMScd} \begin{CD} A @>{f}>> C\\ @V{g}VV @VVV \\ B @>{}>> P = B \coprod_A C \end{CD} $$ where $f$ is a surjection and $g$ is an injection. Does the following claim hold?
Claim: $P$ is a quotient of $B$.
My Thoughts: Define a relation on $B$ as follows: $$ b \sim b' \iff b=g(a), b'=g(a') \ \text{and} \ f(a)=f(a'). $$ To me, this feels like it should be true, but I've no idea how to prove it.
I'd also really appreciate some strong pushout references!
Moveover, suppose that $B$ is itself a quotient space, i.e $B = D/\sim$. Can we thus say that $P$ is a quotient of $D$?
This is not true in general. For instance, if $g$ is an isomorphism, then so is its pushout $C\to P$, so $B\to P$ is a quotient map iff $f$ is a quotient map.
On the other hand, if you assume $f$ is a quotient map, then the map $B\to P$ is also a quotient map (and there is no need to assume $g$ is injective). This is pretty much immediate from the following characterization of quotient maps: a continuous surjection $p:X\to Y$ is a quotient map iff for any $Z$, a function $Y\to Z$ is continuous iff its composition with $p$ is continuous.
In this case, let $p:B\to P$ and $q:C\to P$ be the maps of the pushout diagram and suppose $h:P\to Z$ is any function. Then by the universal property of pushouts, $h$ is continuous iff the compositions $hp$ and $hq$ are both continuous. Since $f$ is a quotient map, $hq$ is continuous iff $hqf$ is continuous. But $hqf=hpg$, so $h$ is continuous iff $hp$ and $hpg$ are both continuous. Since continuity of $hp$ implies continuity of $hpg$, $h$ is continuous iff $hp$ is continuous. Thus $p$ is a quotient map.
If $B$ is additionally a quotient of some other space $D$, then $P$ will be a quotient of $D$ as well, since a composition of two quotient maps is a quotient map.