How Do Sample Spaces and Topologies Intersect in Mathematical Descriptions?

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Discussion Overview

The discussion explores the intersection of sample spaces and topologies within mathematical descriptions, focusing on their structural similarities and differences. Participants consider theoretical implications, potential applications in physics, and the foundational aspects of set theory that underlie both concepts.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants note that both point set topologies and sample spaces involve unions and intersections, but they differ in their foundational structures, such as distance functions and probability density functions.
  • One participant challenges the assertion that all topological spaces have distance functions, clarifying that only metrizable spaces do.
  • Another participant suggests exploring functional analysis as a relevant field connecting these concepts.
  • Some argue that sample spaces can be viewed as measure spaces with a total measure of 1, while topological spaces do not necessarily have an associated measure.
  • A participant raises questions about the nature of the added structures in sample spaces and metric spaces, pondering whether they are arbitrary or if there is an underlying logic connecting them.
  • There is a discussion about the mathematical definitions of probability in sample spaces, including conditional and joint probabilities, and whether these concepts can be simplified to basic multiplicative relationships.
  • Some participants express confusion regarding the relationships between the concepts and seek clarification on the definitions and implications of probability in sample spaces.

Areas of Agreement / Disagreement

Participants express differing views on the foundational aspects of topologies and sample spaces, particularly regarding the necessity of metrics and measures. There is no consensus on the implications of these structures or their interrelations.

Contextual Notes

Participants highlight that set theory serves as a common foundation for both sample spaces and topologies, but the discussion remains unresolved regarding the implications of this relationship and the nature of the added structures.

Who May Find This Useful

This discussion may be of interest to those studying mathematics, particularly in areas related to topology, probability theory, and functional analysis, as well as individuals exploring the mathematical foundations of physics.

Mike2
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Both point set topologies and sample spaces work with unions and intersections of their elements. Point set topologies have distance functions between points, and sample spaces have probability density function at each element. Are there any texts or studies that combine these two disciplines or describe some features of one in terms of the other? Thanks.

I'm wondering if quantum mechanics might be a sample space description and general relativity might be a topological description of the same space.
 
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Mike2 said:
Both point set topologies and sample spaces work with unions and intersections of their elements. Point set topologies have distance functions between points,

no they don't. Only metrizable topological spaces may be equipped with a metric that is compatible with the topology. Very few topological spaces come with a natural metric.

I'm wondering if quantum mechanics might be a sample space description and general relativity might be a topological description of the same space.

You want to look at functional analysis.
 
matt grime said:
You want to look at functional analysis.
Do you have a recommendation on a well written book on the subject at say the beginning/intermediate level? Thanks.
 
Sample spaces are special cases of measure spaces, i.e. essentially they can be viewed as measure spaces with total measure=1. They don't need any topology.

On the other hand topological spaces don't necessarily have any measure associated with them.
 
mathman said:
Sample spaces are special cases of measure spaces, i.e. essentially they can be viewed as measure spaces with total measure=1. They don't need any topology.

On the other hand topological spaces don't necessarily have any measure associated with them.

Yes, I know that. I was struck with the similarity of unions and intersections in both spaces. And I was wondering how artificial was the added struction to turn one into a sample space and the other into a metric space. The sample space seems to have a scalar field, a probability density function, (in the continuous case). The metric seems to be a 2-form field (in the continuous case). Are these added structures completely arbitrary, or is there some underlying logic to them when viewed in a particular way? Certainly in the sample space one can count the elements in an event and compare them to the total number of elements to come up with a probability - this seems like a natural thing to do. Can the same kind of thing be said of a metric space? For example, is a metric just another way of giving a number for a relationship between two elements/points - one is with respect to the whole, the other is with respect to another?

If metrics and probability density functions can naturally be imposed on any space of elements, then constraints in one system might inform the other which might even begin to appear physical. What do you think?
 
Last edited:
Yes, I know that. I was struck with the similarity of unions and intersections in both spaces.

Unions and intersections are operations on sets. Set theory does not require either topology or measure. These make use of added definitions. However, both start with set theory.
 
mathman said:
Unions and intersections are operations on sets. Set theory does not require either topology or measure. These make use of added definitions. However, both start with set theory.

I feel like I'm starting to confuse myself with many issue, and I could use some clarification. Is there a mathematical entity defining the probability of two samples in a sample space occurring? Is this a conditional probability or joint probability? Is it always as simple as the probability of obtain the one sample times the probability of obtaining the second sample?
 
All you are really saying is that they are both based on sets. Just about any mathematics can be based on sets!
 
I feel like I'm starting to confuse myself with many issue, and I could use some clarification. Is there a mathematical entity defining the probability of two samples in a sample space occurring? Is this a conditional probability or joint probability? Is it always as simple as the probability of obtain the one sample times the probability of obtaining the second sample?

P(A&B)=P(A)P(B) when they are independent, otherwise P(B|A)P(A).
 

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