Dual notion of the subspace topology

Question

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram.

Analogously, if $X$ is a topological space, we can consider the final topology induced on $Y$. Does this topology have a name, and does it have any interesting properties? My intuition completely shuts down when I try to think about this object.

I should probably mention, this question occurred to me when I was looking for a "power set topology"

Thanks!

Answer 1

Look at the definition.

$U \subseteq Y$ is open if and only if $f^{-1}(U) \subseteq X$ is open. If $f$ isn't surjective (don't use the word projection), then let $y \in Y$ not in $f(X)$.

Then $f^{-1}(\{y\}) = \varnothing \subseteq X$ which is open. Therefore you get the quotient topology on $f(X)$ and the discrete topology on its complement.