Al-Kalim At-Tayyib

Exam baru habis
Tapi study masih study...
Orang lain exam pun tak habis, tapi dah kene start kerja...


Introduction to Convergence Theory

Recall that a function f is differentiable iff it is differentiable at all points in its domain
A priori knowledge of generalized derivatives such as the frechet derivative is not necessary

From undergraduate calculus,
Let f : X -> Y be differentiable, then f is continuous
The converse is not true.
Note X and Y are usually taken to be the reals in undergrad calc, but this holds for more general spaces as well

One might wonder however, could we not find a topology such that differentiability equals continuity?
This was answered negatively by ...(forgot who, this part kene review balik after this)

However it turns out that we can generalize topological spaces such that differentiability IS continuity, such spaces are called convergence spaces(well at least that's what I think it's called)

A relation between filters on a set X and the elements of X, denoted by
x in lim F
is called a convergence on X
provided that G contains F implies lim G contains lim F
and that the principal ultrafilter of every x is in this relation with x.

Each topological space defines a convergence space on the same set, x in lim F whenever F contains every open set containing x.
Such a convergence is said to be topological.

This sounds like a whole lot of mumbo jumbo to me atm.
Unfortunately since my uni does not offer undergraduate topology atm for some reason, I'll have to read up on some stuff first

Next up, Filters(generalization of nets)
Will probly only touch a few bits here and there, while putting stuff like filter bases and whatnot into an appendix