Final answer:
In general, a continuous map will still be continuous if the topologies on X or Y are changed to finer or coarser ones.
Step-by-step explanation:
In general, if the topology on X is changed to a finer one or the topology on Y is changed to a coarser one, the map f:X→Y will still be continuous.
If the topology on X is changed to a finer one, it means that more open sets are added to X. Since every open set in the new topology is already open in the original topology, the preimage of any open set in Y under the map f will still be open in the new topology.
If the topology on Y is changed to a coarser one, it means that some open sets in Y are removed. Since the preimage of any open set in Y under the map f is still open in the original topology, it will also be open in the new topology.