Intereting Posts

Theorem about positive matrices
If $p$ divides $a^n$, how to prove/disprove that $p^n$ divides $a^n$?
$f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors
Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
The pigeonhole principle – how to solve questions like that?
Uniform Continuity of $f(x)=x^3$
Problem understanding how to compute fundamental group of connected sum of torus
Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
Discontinuous functions with closed graphs
Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$
Elegant solution to $\lim\limits_{n \to \infty}{{a} – 1)]}$
Finding total after percentage has been used?
Comparing $2013!$ and $1007^{2013}$
Why is polynomial regression considered a kind of linear regression?
What is the best way to develop Mathematical intuition?

I have learned point-set topology using filters. Now I do functional analysis where we are using nets to do topological stuff. Therefore I search an introductory text on nets that is suitable for this purpose, i.e. to lay the foundations for usage in FA and maybe that the text assumes some knowledge in point-set topology, I don’t want to start from scratch.

- What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?
- Dense subset of $C(X)$
- When is the topological closure of an equivalence relation automatically an equivalence relation?
- Existence of a continuous function which does not achieve a maximum.
- If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected?
- Proving that if a set is both open and closed then it is equal to the real numbers
- Is a compact simplicial complex necessarily finite?
- $|f(x)-f(y)|\le(x-y)^2$ without gaplessness
- Relation between convergence class and convergence space
- Given any base for a second countable space, is every open set the countable union of basic open sets?

One of the very best references that I’ve seen is a PDF, *Translating Between Nets and Filters*, by Saitulaa Naranong that can still be found here. (I’m aware of one typo: $\Phi$ and $\Psi$ have been interchanged in the displayed implication at the top of page $11$. The one-sentence paragraph two lines down (‘In other words …’) is correct.)

Two sources:

- Kelley,
*General Topology*, which popularized nets and the terminology, and - Pete Clark, Convergence, a paper developing the theory of sequences, filters and nets, and proving implications and equivalences between them.

A caveat to my answer: I don’t know functional analysis, so this is just what has helped me in understanding point set topology better. I find nets build intuition in point set topology.

In no particular order:

- Try the section in Engelking’s
*General Topology*starting on p. 49. - Munkres’s
*Topology*has a little section on nets with some good exercises. - Willard’s
*General Topology*chapter 4 is good.

I recommend learning the canonical way to translate nets into filters, and vice-versa. If you still want more practice, I recommend going through some of the proofs in point set topology (e.g. tube lemma) in the context of nets, and then the corresponding one in terms of filters. It’s fun!

I don’t know a reference that explains nets for functional analysis, but here is an introductory book on nets:

Limits – A New Approach to Real Analysis by Alan F. Beardon.

- Gradient of vector field in spherical coordinates
- Galois theory: splitting field of cubic as a vector space
- How prove binomial cofficients $\sum_{k=0}^{}(-1)^k\binom{n+1}{k}\binom{2n-3k}{n}=\sum_{k=}^n\binom{n+1}{k}\binom{k}{n-k}$
- Any ring is integral over the subring of invariants under a finite group action
- Radius of a cyclic quadrilateral given diagonals
- Proof of trigonometric identity $\sin(A+B)=\sin A\cos B + \cos A\sin B$
- Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
- Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
- Splitting of primes in an $S_3$ extension
- Achilles and the tortoise paradox?
- Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$
- There always exists a sequence of consecutive composite integers of length $n$ for all $n$.
- Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
- Why are very large prime numbers important in cryptography?
- Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$