# Questions tagged [nets]

A net is a generalization of a sequence where a directed set is used as the index set instead of positive integers. Convergence of nets can be defined in a similar way as convergence of sequences. Convergent nets in a topological space uniquely determine its topology.

261 questions
Filter by
Sorted by
Tagged with
13 views

29 views

### Munkres supplementary exercises chapter 3(nets) question 10

Prove the following. X is compact if and only if every net in X has a convergent subnet. I am stuck in proving the (<=) side. https://dbfin.com/topology/munkres/chapter-3/supplementary-exercises-...
49 views

### Visualizing nets in a topological space

What is the best way one can visualize the concept of a net or a subnet in a given topological space $X$? What is the intuition that makes sence when thinking about the definitions of convergence and ...
23 views

### Lim sup nets proof, is it valid?

I just get really nervous around lim sup and nets for some reason... Please help! Problem I was given: Suppose $(x_i)_{i\in I} \subset B(\mathcal{H})$ converges to $x \in B(\mathcal{H})$ in the SOT (...
45 views

### Closedness of a set by using convergent nets

I am currently studying about nets, so it is all new to me. There is one thing that I could not find anywhere, so I try ask here. (If you know a reference, please let me know.) Let $X$ be a ...
42 views

53 views

### Construct a net on the unit ball in $\ell^1 (\mathbb N)$ weakly converging to zero

Recall that $\ell^1 (\mathbb N)$ has Schur's property, that every weakly convergent sequence is strongly convergent, i.e. convergent with respect to the norm $||\cdot||_1$. However, the weak topology ...
41 views

### Net based on a filter

Assume a topological space X, t. I read a proposition in my textbook: A filter A $\to$ y if and only if every net {$s_a$}, a $\in$ A, based on A also converges to y. What are the net elements $s_A$ in ...
32 views

### What exactly is the relationship between weak* convergence and sequential convergence?

I am studying Banach algebras, and the theory frequently makes use of the weak* topology which I am not very familiar with. As I understand it, the weak* topology is defined as follows: let $X$ a ...
55 views

### Question on metric spaces and nets

I am self studying Topology from Gemignani's Elementary Topology. Here's the question which I am trying to prove (Exercise 2 on page 127): Let $X,D$ be a metric space and $\{ s_i \}, i \in I$ be a ...
54 views

### Definition of subsequence of a net

I'm self studying Topology from Michael Gemignani's Elementary Topology. The author asks the following question (Exercise 2 on page 127): Suppose $X,D$ is a metric space and $\{ s_i \} , i \in I$, ...
41 views

### Almost Everywhere Convergence on Lebesgue Measure and Topology

I am reading the book "Z. Semadeni, Banach Spaces of Continuous Functions". At Definition 3.6.11, he defines a generic notion of convergence as follows: By "upward filtering ordered sets" he means a "...
62 views

38 views

23 views

### An increasing convergent net $(x_{\alpha})$ of real numbers is bounded by its limit.

Let $(x_{\alpha})$ be an increasing net, i.e. $\alpha\leq\beta\implies x_{\alpha}\leq x_{\beta}$, in $\mathbb{R}$ that converges to $x$. Can we conclude that $x_{\alpha}\leq x$ for all $\alpha$? In ...
45 views

### “limit” vs. “limit point” of a sequence in a topological space

Let $(E,\tau)$ be a topological space and $(x_n)_{n\in\mathbb N}\subseteq E$. I'm highly confused by the notion of a limit point $x\in E$ of $(x_n)_{n\in\mathbb N}$. If $\tau$ is induced by a ...
15 views

### Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
64 views

149 views

### weak star and strong convergence of net in Banach spaces

In Banach spaces, the following result is well-known: (1) Let $X$ be a Banach space. Let $\{x_n\}\subset X$ and $\{x^*_n\}\subset X^*$ be such that $x_n \rightarrow x$ (convergence with respect to ...
I have a question about nets. I consider an infinite directed set $(A, \prec_A)$ and a net $\{x_a\}_{a \in A}$ in a compact topological space $X$. Then, it admits a convergent subnet (I consider ...
Problem. If $(x_i)$ is a net converging to a point $x$, show that $(x_i)$ converges to $x$ along any nonprincipal ultrafilter. I will define these things below. A directed set is a poset $(I,\leq)$ ...