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.

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Product topology and nets

I just learned about nets, but I'm confused about when to use them. More precisely when combined with products. Let $(X_i)_{i\in I}$ be a family of topological spaces and consider $X = \prod_{i \in I} ...
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Cofinal subset of a directed set such that a given limit is infinite

Let $D$ be the directed set consisting of all functions $f: \mathbb N \to \mathbb N$ with the standard preorder $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \mathbb N$. Let $(n_d)_{d\in D}$ be a net ...
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If net $f$ converges to $x$ and $y$ is an accumulation point of $f$ then can it be proved that $f$ converges to $y$?

Let $\mathcal J$ be a directed set, let $X$ be a topological space and let $f:\mathcal J\to X$ be a function. Then $f$ is a so-called net on $X$. It can converge to elements of $X$ and can have ...
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Does a net converging to a point $x$ have a countable subset $\{x_n\}_n$ such that it converges to $x$ in a sequentially compact space?

I was thinking for example, in functional analysis, we can talk about a Banach space $E$ and it's dual space $E^{*}$ with it's $w^{*}$-topology. In the special case that $E$ is infinite dimensional ...
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Closures and nets in topological spaces

Suppose $(X,\tau)$ is a topological space and $A\subset X$. Let $\overline{A}$ denote the closure of $A$ in $X$. Suppose $x\in A$. Then there exists a net $\langle x_\delta\rangle_{\delta\in\Delta}$ ...
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How to show that if x is a cluster point of a filter then it a cluster point of each of its associated or derived net.

Am stuck with this problem. However I am able to show the other way round. Please help me wih it.
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Show that any compact uniform space (compact for the topology of the uniformity) is complete

My efforts: Let the uniform space be $(S,\mathcal{U})$. For a Cauchy net {$x_\alpha$}, the collection of all $B_\gamma$ = {$x_\alpha:\alpha\geq\gamma$}, $\gamma\in I$, is a filter base that extends to ...
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Countability and convergence in topological spaces

The main Idea here is that sequence convergence in not first countable spaces is not enough to explain closure and continuity. I want to discuss this with two examples: (1) Lets have $\mathbb{R}^{\...
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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-...
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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 ...
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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 (...
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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 ...
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Definition of convergence of a net

Let $(x_\alpha)_{\alpha\in I}$ be a set in $X$, where it is used $(I,\preceq)$ as a directed set. Which one of these definitions are correct, when we learn about the convergence of a net $(x_\alpha)_{\...
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Wordings of definition of an accumulation point of a net

Let $O(x)$ denote the set of open sets containing a point $x$. I read in a definition, A point $x$ in a topological space $(M,\tau)$ is called an accumulation point of the net $(x_i)_{i\in I}$ in $M$ ...
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What if: Compact operators defined as limits of nets of finite-rank operators instead of norm closure?

I am interested in this question from my operator theory study, but I am terrible at seeing the difference between nets and sequences... Can anyone share an idea? Especially I prefer to visualize ...
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Does the following net define a finitely additive probability measure?

Let $\mathcal X$ be a set, and let $\mathcal F$ be the set of all finite subsets of $\mathcal X$ directed by subset inclusion. For each finite set $F \in \mathcal F$, let $\mu_F$ be the probability ...
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Question about strongly convergent nets.

Consider the following theorem in Murphy's book "$C^*$-algebras and operator theory": Why do we need to truncate the net in order to conclude that $(u_\lambda)_{\lambda}$ is bounded below? ...
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Question about cluster point and subsequence on topological space.

First, take some definition: Given a topological space and net, defined on : . We say that x is cluster point of net, if for every open set and for every , there exist , such that . We know that ...
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A function is integrable if and only if the nets of lower sum and upper sum converge to the same number.

Definition 1 If $\mathscr{P}$ is the set of all partition of a rectangle $Q$ of $\Bbb{R}^n$ then we say that $$ P_1\preccurlyeq P_2\,\Leftrightarrow\,\text{any point of} \,P_1\,\text{is a point of}\,...
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closure of nets in von-Neumann algebras

I found the following assertion in my class notes, but i did not understood why doe's it holds: "Suppose that in an infinite-dimensional von Neumann algebra there is an increasing net $\{p_j\}$ ...
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Show that in [0,1] with its usual topology there exists a net having no convergent strict subnet

I only have difficulties in the final step: Show that {$x_y:y\in I$} has no convergent strict subnet. My efforts: With the construction, $I$ is a minimal uncountable well-ordered set. Thus it has the ...
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An ultranet $x_\lambda$ is frequentely in $Y$ if and only if it is residually too.

Definition If $x_\lambda$ is a net from a directed set $\Lambda$ into $X$ and if $Y$ is a subset of $X$ then we say that $x_\lambda$ is redisually in $Y$ if there exsit $\lambda_0\in\Lambda$ such that ...
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isomorphism filter and nets

Given a filterbasis, by picking up an point out of every element of this filterbasis, and considering the ordering on the elements of the filterbasis $u\leq v$ iff $v \subset u$ (subquestion why is ...
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Show if $f_x \rightarrow \eta \,\,\,$ and $g_x \rightarrow \zeta$ so $f_x+g_x \rightarrow \eta + \zeta$

Let $(f_x)$ and $(g_x)$ be two nets on a directed set $X$. Show if $f_x \rightarrow \eta \,\,\,$ and $g_x \rightarrow \zeta$ so $f_x+g_x \rightarrow \eta + \zeta$ For $(f_x)$ holds: $$\forall \...
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Show if $f_x\to \eta,\;\,g_x\to \zeta$ and if $\forall \,x\succ x_0:f_x \le g_x$, then $\eta \le \zeta$.

Show if $f_x\to \eta$, $\; g_x\to \zeta$ and if $\forall \,x\succ x_0:f_x \le g_x$, then $\eta \le \zeta$. $(g_x)$ and $(f_x)$ both are nets on the directed set $(X,\succ)$ with $X\subset \mathbb{R}$....
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Define a filter base according to a net

Let $\{x_i\}_{i\in I}$ be a net in a topological space. Define a filter base $\mathcal{F}$ such that for all $x$, $\mathcal{F}\rightarrow x$ if and only if $x_i\rightarrow x$. My efforts: Define $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\{ s_i \} , i \in I$, ...
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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 "...
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Net compactness and relative compactness

I'm trying to understand the relation between the following conditions. I will assume that $X$ is a Hausdorff topological space and $A \subset X$. $\overline{A}$ is compact; Every net $\{x_{\lambda}\}...
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A challenging problem related to characterisation on BV-functions.

I was trying to find the answer to the following question: Let $f$ be a BV-function on $[a,b]$,with total variation $V_a^b(f)$.Then does it imply that for any sequence $\{P_n\}$ of partitions of $[a,...
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Net (Mooore - Smith) Convergence question

Give topological space $(X,\tau)$, $E \subset X$, $\{x_{\alpha}\} \subset E, x\in E$ and $(E,\tau_{E})$ is subset topological space of $(X,\tau)$. I have proved that: If $x_{\alpha} \xrightarrow{(E,\...
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Cluster point of a net is a limit of a subnet

A point $c\in X$ is a cluster point of the net $(x_d)_{d\in D}$ if, for every neighborhood $U$ of $c$ and for any $d_0\in D$ there exists $d\ge d_0$ such that $x_d\in U$. In the other words, $x_d$ is ...
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Limit superior is a cluster point of a net

Let $(x_d)_{d\in D}$ be a net net of real numbers. Limit superior of a net is defined as $$\limsup x_d = \lim_{d\in D} \sup_{e\ge d} x_e = \inf_{d\in D} \sup_{e\ge d} x_e.$$ See, for example, Limsups ...
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A question regarding nets of matrices

I am trying to understand the following theorem from this paper I understand the first part of the proof. Basically we are adding in $F_\alpha$ all matrices with entries from $\{-2, -2+\frac{\sqrt{\...
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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 ...
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“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 ...
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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 (...
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Understanding $\epsilon-$net arguments for random matrix bounds

I'm reading this book to learn about $\epsilon$-net arguments for random matrix concentration bounds. In Section 4.2.2, a high probability upper bound is computed for the operator norm of matrix $A \...
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Is a Banach space a complete topological vector space?

A net $(v_\alpha)_{\alpha\in I}$ in a topological vector space (=TVR) $(V, \mathcal{T})$ is called Cauchy if $$\forall U \in \mathcal{V}_V(0): \exists \alpha_0 \in I : \forall \alpha, \beta \geq \...
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Nets, Cluster points, and closure: Showing a space is compact if every net has a cluster point

I'm trying to understand the following proof that a space $X$ is compact if and only if every net has a cluster point. I have a specific confusion with how cluster points relate to closure which is ...
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Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

Suppose I have an equation in some reflexive separable Banach space $X$: $$Au=f$$ for given data $f$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
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How to interpret the sum “$\sum_{k,l\in\mathbb{N}}$”?

Suppose that we have a collection of (non-negative) real numbers $\{a_{k,l}:k,l\in\mathbb{N}\}$, then what does the series/sum/limit $\sum_{k,l\geq1}a_{k,l}$ actually mean? For example: Does it mean ...
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If $(a_{i}(k,l))_{i\in I}$ are nets in $[0,\infty)$ for all $k,l\in\mathbb{N}$, conv. to $0$, then $\sum_{k,l\in\mathbb{N}}a_{i}(k,l)/2^{k+l}\to0$.

Suppose that for each $k,l\in\mathbb{N}$ we have a net $(a_{i}(k,l))_{i\in I}$ in $[0,\infty)$ that converges $0$ as "$i\to\infty$". Also, there is a constant $C$ such that $a_{i}(k,l)\leq C$ for all $...
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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 ...
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20 views

Subnet with an infinite directed set

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 ...
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108 views

Ultralimit extends limit of nets?

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 $(I,\leq)$ ...

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