Algebraic and topological sets

In summary, a countably infinite set can be an algebra, while an uncountably infinite set can be a topology. However, both types of sets can have interesting and useful properties in the context of topological and algebraic structures.
  • #1
SW VandeCarr
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81
Is it an oversimplification to say a countably infinite set is an algebra while an uncountably infinite set is a topology?
 
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  • #2
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.
 
  • #3
Hurkyl said:
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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  • #4
SW VandeCarr said:
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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1. What is the difference between algebraic and topological sets?

Algebraic sets are defined using algebraic equations or polynomials, while topological sets are defined using concepts of continuity and open/closed sets. In other words, algebraic sets have a more geometric approach, while topological sets have a more analytical approach.

2. How are algebraic and topological sets used in mathematics?

Algebraic and topological sets are used in a wide variety of mathematical fields, including algebraic geometry, topology, and mathematical analysis. They provide a powerful framework for studying and understanding the structure of mathematical objects such as curves, surfaces, and higher-dimensional spaces.

3. Can you give an example of an algebraic and topological set?

An example of an algebraic set would be the set of points on a plane that satisfy the equation x^2 + y^2 = 1, which is the equation of a circle. An example of a topological set would be the set of all real numbers between 0 and 1, which forms an open interval in the real line.

4. How are algebraic and topological sets related to each other?

Algebraic sets can be studied using topological methods, such as the concept of Zariski topology. In fact, algebraic geometry is often seen as the study of the interplay between algebraic and topological structures. Additionally, many topological spaces can be described using algebraic equations, leading to connections between the two fields.

5. Are there any real-world applications of algebraic and topological sets?

Yes, there are many real-world applications of algebraic and topological sets, particularly in fields such as computer graphics, robotics, and machine learning. These sets are used to model and understand complex structures and patterns in real-world data, leading to advancements in technology and science.

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