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There isn't really such a thing as a 'continuous topological space'. Technically speaking, continuity is a property of functions between topological spaces. I t
by m3ndax 3y ago
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 3y agoThe 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 3y agoConnected or complete then.