Algebraic and topological sets

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Discussion Overview

The discussion revolves around the relationship between countably infinite sets and algebras, as well as uncountably infinite sets and topologies. Participants explore definitions and examples from algebra and topology, questioning whether the categorization of sets as algebras or topologies based on their cardinality is an oversimplification.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants suggest that categorizing countably infinite sets as algebras and uncountably infinite sets as topologies may be an oversimplification.
  • Others point out that there are countably infinite topological spaces and uncountably infinite algebras, indicating that the relationship is not straightforward.
  • One participant references definitions of topological sets and algebraic sets, noting that topological sets consist of open sets while algebraic sets consist of closed sets.
  • There is a mention of confusion regarding the distinction between closed and open sets in relation to countable and uncountable sets, with an example of the closed interval [0,1] being described as "countable" due to its endpoints.
  • Another participant elaborates on the nature of topological spaces, providing examples of different topologies that can exist for a given set.

Areas of Agreement / Disagreement

Participants express differing views on the classification of sets as algebras or topologies based on cardinality, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

There are unresolved aspects regarding the definitions of topological and algebraic sets, as well as the implications of cardinality on these classifications. The discussion also highlights potential confusion between the concepts of closed and open sets.

SW VandeCarr
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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?
 
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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.
 
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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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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