Does this topology have a name? Suppose that $(X,\tau)$ is a topological space, $A\subset X$ is an open  subset. Define a new topology, $\tau_A$, on $X$,
$$
\tau_A=\{U\subset X: U\cap A\in \tau\}.  
$$
It is easy to see that $\tau_A$ is again a topology on $X$.
Question. Is there a standard name for the topology $\tau_A$?
Edit. The suggested (in a comment) list of topology books has nothing to do with my question.
 A: I don't know if $\tau_A$ has a standard name, but it is produced by a standard construction.
Let $i : A \to X$ be the inclusion. Then $\tau_A$ is the the finest topology on $X$ which makes $i$ continuous with respect to the subspace topology on $A$. In other words, $\tau_A$ is the final topology induced by $\{i\}$.
This is sort of a "backwards-forwards" procedure. Starting with $(X,\tau)$, we produce $\tau_A$ as follows:

*

*Let $\sigma$ be the initial topology on $A$ induced by the inclusion $A \to X$ and the topology $\tau$ on $X$. (I.e. we use $i$ to induce a topology on $A$ from the topology on $X$)

*Let $\tau_A$ be the final topology on $X$ induced by the inclusion $A \to X$ and the topology $\sigma$ on $A$. (I.e. we use $i$ to induce a topology on $X$ from the topology on $A$)

A: One could call it the topology generated by $A$, with “generated” being in the same sense as “compactly” generated.
According to Wikipedia we might also call such a topology the topology coherent with $A$. Well, if $\tau_A=\tau$, we say $(X,\tau)$ is coherent with the subspace $A$.
