Fully Developed Areas in Mathematics

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Discussion Overview

The discussion revolves around the status of "point-set theory" and "vector analysis" in mathematics, specifically questioning whether these fields are fully developed or if ongoing research is still relevant. The scope includes theoretical aspects and the relationship between different mathematical disciplines.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants suggest that point-set theory was considered largely complete over 40 years ago, yet acknowledge that new ideas could still emerge.
  • There is a question about the distinction between point-set theory and set theory, with some asserting that they are not the same, as point-set theory is part of topology while set theory focuses on logic fundamentals.
  • One participant claims that point-set theory has been significantly developed but emphasizes that it is not accurate to say that no further research is needed, particularly in general topology.
  • Vector analysis is described as a classical subject that has been integrated into broader fields such as differential geometry and differential topology, with ongoing research in these areas.

Areas of Agreement / Disagreement

Participants express differing views on whether point-set theory and vector analysis are fully developed. While some believe that point-set theory has reached a level of completion, others argue that research is still viable. The relationship between point-set theory and set theory is also contested, indicating a lack of consensus.

Contextual Notes

There are unresolved questions regarding the definitions and scope of point-set theory and vector analysis, as well as the implications of their integration into other mathematical fields. The discussion reflects varying perspectives on the state of research in these areas.

RJ Emery
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Is there in mathematics a field of study known as "point-set theory," and is this an area that has been fully developed that no further research is needed or being performed?

Can the same be said for vector analysis?
 
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point set theory was considered pretty much over 40 years ago or more, but someone could always think of something new.
 
mathwonk said:
point set theory was considered pretty much over 40 years ago or more, but someone could always think of something new.
How does "point set theory" differ from "set theory"? Or are the two the same?

What about vector analysis? Is that too an area where research is largely over?
 
Vector calculus, in modern mathematics, is subsumed into differential geometry. There is ongoing research into many aspects of differential geometry (for example, symplectic geometry, flows, etc.).

Point set theory is not particularly related to set theory. Point set theory is part of topology. Set theory deals with the fundamentals of logic.
 
Last edited:
RJ Emery said:
Is there in mathematics a field of study known as "point-set theory," and is this an area that has been fully developed that no further research is needed or being performed?

Can the same be said for vector analysis?

Point set theory is simply topology. The point set theory of real numbers is the topology of the real numbers.
Point set theory has been much developed. Yet no one can say that no further research is needed - you may very well be interested in research in general topology.

Vector analysis is a very classical subject. It has now been subsumed into a larger set of classes - differential geometry, differential topology etc etc.
 

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