characterization of subspace topology
Let denote the subspace topology on and denote the inclusion map.
Suppose is a family of topologies on such that each inclusion map is continuous. Let be the intersection . Observe that is also a topology on . Let be open in . By continuity of , the set is open in each ; consequently, is also in . This shows that there is a weakest topology on making inclusion continuous.
We claim that any topology strictly weaker than fails to make the inclusion map continuous. To see this, suppose is a topology on . Let be a set open in but not in . By the definition of subspace topology, for some open set in . But , which was specifically chosen not to be in . Hence does not make the inclusion map continuous. This completes the proof. ∎