[nyrikki]

The classic middle-thirds Cantor Set being a topologically set is one of the easiest counter examples to the above misconception that the sets need to be continuous themselves.

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.