Name for continuous maps satisfying $\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U)$ I have recently come across particularly kind of continuous maps $f 
\colon X \to Y$ between topological spaces with the property that 
$$
\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U),
$$
for all $U \in \wp(X)$. 
If $X$ is $T_1$ then this condition implies that $f$ will be an injection. However, the spaces I consider are not necessarily $T_1$. Therefore, I would like to know if these kinds of maps have a name or if they have been considered before.  

Look at the fibers. Your condition implies that $f^{-1}(y) ⊆ cl(\{x\})$ for every $x ∈ f^{-1}(y)$, i.e. each fiber is indiscrete. So, such mapping is injective even if $X$ is $T_0$. Conversely, if each fiber is indiscrete, then the closure contains the “fiber-closure”.
Every continuous map is a unique composition of a quotient map and an injective continuous map. This condition is equivalent to the condition that the $T_0$ quotient factorizes through the quotient part of the original map.
