In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity. Your statement doesn’t even make sense. It’s maps between them that can be continuous.
In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.
Being able to define a neighborhood or a concept of closeness is required, but the concept of distance is not required.
If you can define a distance a topological space is a metric space
If it is locally euclidean it may be a manifold.
Really the union and finite intersection of subsets is the formal way of showing something is a topological space. Too har do describe here but that is where the concept of continuity arises.
