I don't understand the following logic:
If $f: X \rightarrow Y$ is a function from a topological space $X$ to a topological space $Y$, and $f^{-1}(U)$ is an open set of $X$, where $U \subset Y$, and $f$ is continuous, then $U$ is an open set of $Y$.
Usually, how it proceeds is that we are first given that $f$ is continuous, then that $U \subset Y$ is an open set, and we conclude that $f^{-1}(U)$ is an open set of $X$. But we are doing the logic backwards here, and I don't understand how we can conclude that $U$ is an open set of $Y$.
Would somebody be able to explain how we reach this conclusion?
Thanks.
This phenomenon that we "test" openness in $Y$ by looking at the inverse image in $X$ happens exactly when $f$ is a quotient map, or alternatively put, that $Y$ has the quotient (or "final") topology w.r.t. $f$. See wikipedia or any good topology text book. So it is a special circumstance.