Let $ X $ be a Banach space, and $Y$ a closed subspace, and let the quotient map be $ q : X \to X/Y$.
Now, if $X$ and $X/Y$ are given the norm topology (where the norm on $X/Y$ is $\|x+Y\| = d(x,Y)$), then we know the induced quotient topology and the norm topology on $X/Y$ coincide. The same is true if $X$ and $X/Y$ are given the weak topology.
Does something like this hold in general? That is, if $X$ and $X/Y$ are given the same "kind" of topology, then the topology on $X/Y$ coincides with the induced quotient topology.
I appreciate this might be vague, because I don't know how to formulate precisely what it means for $X$ and $X/Y$ to have the same "kind" of topology, but any further information helps.