Let (X,T), (Y, U) topological spaces, $f: X \to Y$ is a continuous function. Is the preimage of a basic open set in Y always a basic open set in X? Is the preimage of a subbasic open set in Y always a subbaisc open set in X?
Here basic open set is an element of a basis, subbasic open set an element of a subbasis.
Best Answer
No. For an example, take $X:=Y:=\Bbb R^2$ with the same Euclidean topology, but with the open disks as a basis for $X$ and the open squares as a basis for $Y$, and take $f:={\rm id}$.