[anon291]

Perhaps the confusion is that I should have said topological spaces can be continuous. There are discrete topological spaces. Topologies (which I believe is typically used to refer to the collection of open sets in a topological space) are not functions or relations themselves, so I'm not sure a useful notion of continuity applies there, but if I'm wrong, please inform.

[m3ndax]

There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I think you're being tempted to use the terms continuous and discrete in a more colloquial sense mapping more to uncountable vs countable/countable and finite perhaps. But yeah, you really wouldn't use the term continuous to describe a topological space or a topology.

[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.

[anon291]

Connected or complete then.