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

Staff Emeritus
Gold Member
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.

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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).

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