I'm going through a fairly involved proof in Algebraic Topology, and am stumbling at the last hurdle because my point-set topology is rusty.
Suppose I have a map $f : Z \to Y$, where $Y$ and $Z$ are topological spaces. If I've shown that $f^{-1}(A)$, where $A$ is any open set in a basis for the topology on $Y$, contains a set open in $Z$, does it follow that $f$ is continuous?
Thanks