Fully Developed Areas in Mathematics

[RJ Emery]

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?

[mathwonk]

point set theory was considered pretty much over 40 years ago or more, but someone could always think of something new.

[RJ Emery]

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?

[Data]

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.

[sauravbhaumik]

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.