Fully Developed Areas in Mathematics

In summary: Vector analysis is still very much an active area of research.In summary, point set theory is a field of study in topology that has been developed over the past 40 years, but there is always potential for new discoveries. It should not be confused with set theory, which is a separate field. Similarly, vector analysis has been incorporated into other areas of mathematics such as differential geometry, but it is still an active area of research.
  • #1
RJ Emery
113
6
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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  • #2
point set theory was considered pretty much over 40 years ago or more, but someone could always think of something new.
 
  • #3
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?
 
  • #4
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.
 
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  • #5
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.
 

1. What is the definition of a "fully developed area" in mathematics?

A fully developed area in mathematics refers to a field or subfield that has reached a high level of complexity and sophistication, with a significant amount of research and applications. These areas are typically well-established and have a large body of knowledge and techniques.

2. How do mathematicians determine when an area is considered "fully developed"?

The determination of a fully developed area in mathematics is subjective and can vary among mathematicians. However, some factors that may contribute to this determination include the number and quality of research publications, the number of active mathematicians working in the area, and the level of complexity and depth of the concepts and techniques involved.

3. What are some examples of fully developed areas in mathematics?

Some examples of fully developed areas in mathematics include calculus, linear algebra, differential equations, abstract algebra, and topology. These areas have a wide range of applications in various fields, and there is a wealth of research and literature available on these topics.

4. Are fully developed areas in mathematics constantly evolving?

Yes, fully developed areas in mathematics are constantly evolving as new research is conducted and new techniques and applications are discovered. This is why many mathematicians continue to work in these areas, even though they are considered well-established.

5. How can studying fully developed areas in mathematics benefit other fields?

Studying fully developed areas in mathematics can benefit other fields by providing powerful tools and techniques for problem-solving and analysis. Many areas of science, engineering, and economics rely heavily on mathematics, and advances in fully developed areas can lead to new and innovative applications in these fields.

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