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Is it an oversimplification to say a countably infinite set is an algebra while an uncountably infinite set is a topology?
The definition of a topological set in the following refers to the set T which consists (only?) of open sets.Yes.
It might make more sense with some surrounding context, but I do feel compelled to point out that some of my favorite topological spaces have countably many points, and some of my favorite algebras have uncountably many points.
The set T defines which sets are open. A given set can have many possible topologies. The open sets in Euclidean space have a metric topology, which is a topology T generated by all balls B(y, r) such that B(y, r) = {y | d(y, x) < r} where d is the Euclidean metric.The definition of a topological set in the following refers to the set T which consists (only?) of open sets.
http://knowledgerush.com/kr/encyclopedia/Topological_space/