Algebraic and topological sets

[SW VandeCarr]

Is it an oversimplification to say a countably infinite set is an algebra while an uncountably infinite set is a topology?

 

[Hurkyl]

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.

 

[SW VandeCarr]

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 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/

On the other hand the following states that algebraic sets consist of closed sets.

http://mathworld.wolfram.com/AlgebraicSet.html

Perhaps I'm confusing closed and open sets with countable and uncountable sets. For example, the closed interval [0,1] is "countable" because 0 and 1 are included in the set.

 

[slider142]

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/

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.
A simple topological space is the pair consisting of the set 2 = {a, b} with the topology T = {{a}, 2}, where only the singleton {a} and the entire set 2 is open (as well as the trivial empty set). Other possible topologies for this set include T = {{a}, {b}, 2} which is the discrete topology (every discrete point is open) and T = 2 which is the concrete or indiscrete topology ((2, T) is as impenetrable as a slab of concrete, the only non-empty open set is 2 itself).