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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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Why is sequences enough in the definition of a normal family

In the definition of a normal family in complex analysis, we are concerned with a sequence of functions having a subsequence uniformly converging to a function on compact subsets of an open domain. ...
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If $(y_\lambda)_{\lambda \in \Lambda}$ is a subnet of $(x_\alpha)_{\alpha \in A}$ and if $A$ is a complete lattice, is $\Lambda$ a complete lattice?

Let $X$ be a set, $(x_\alpha)_{\alpha \in A}$ be a net in $X$ and $(y_\lambda)_{\lambda \in \Lambda}$ be a subnet of $(x_\alpha)_{\alpha \in A}$. If $A$ is a complete lattice (i.e. not only a directed ...
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How to construct the nets using the coordinate projection on the cube?

I am reading a paper, but I don't quite understand how they construct the nets for a set. I hope someone can explain why this method works. I have previously encountered how to construct nets on a ...
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how to understand the epsilon-nets are constructed in the followsing set about the $n$-dimentisonal vector?

I am reading an article about constructing nets on a set, but I do not fully understand how the epsilon-nets are constructed. The general idea is to partition the size of the coordinates of a vector ...
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Non-overlapping subnets always exist?

Suppose a net $(\alpha_i)_{i\in I}$, where $I$ is a directed set with no maximal element and $X$ a topological space, has no duplicate values. That is, for $i\neq j\in I$, $\alpha_i\neq \alpha_j$. Is ...
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Convergence of a net and its associated filter

I am reading "Topology: An Introduction" by Waldmann and I am trying to prove the result (iv) in Proposition 4.2.6: A net converges to $p$ $\iff$ its associated filter converges to $p$. I ...
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How are $\epsilon$-nets (as in centers of $\epsilon$-ball covers) related to nets (as in topology)?

Here are two definitions that I have encountered: The first, corresponding to this Wikipedia page, is the following. Definition.$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$....
Ardy's user avatar
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Testing points on the closure of a union of increasing subsets

Let $X$ be a topological space. Let $(S_\lambda)_{\lambda\in\Lambda}$ be a net of closed subspaces of $X$, and suppose that for all $\lambda\le\mu$, $S_\lambda\subseteq S_\mu$. Let now $$ x\in \mathrm{...
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How to formulate the condition $\forall t \in [0,1]: p_t(x_n-x) \underset{n \rightarrow \infty}{\rightarrow} 0 $ in the sense of net convergence

Consider the following: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real valued functions $[0,1] \rightarrow \mathbb{R}$. And $\mathcal{P}:={p_t:t \in [0,1]}$ a family of seminorms such that $p_t(...
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Filtration of separable Hilbert spaces

Let $\mathcal{H}$ be a separable Hilbert space. Let $(\mathcal{S}_\alpha)_{\alpha\in\Lambda}$ be a decreasing net of closed subspaces of $\mathcal{H}$, i.e. such that for each $\alpha,\beta\in\Lambda$,...
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Cofinality in the definition of subnets

Why is cofinality required in the definition of a subnet? My professor mentioned that there are certain cardinality-related reasons for this requirement. Does this have anything to do with the ...
Iskander's user avatar
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When a net of positive real numbers converges?

Let $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ be a two sets of non-negative real numbers. If $a_i\leq b_i$ for every $i$, and $\sum_{i\in I}b_i=1$, then when $\sum_{i\in I}a_i$ converges? If $I$ ...
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Existence of a diagonal subsequence in a Fréchet-Urysohn space

Let $X$ be a Fréchet-Urysohn space (i.e., for all $A\subseteq X$, the closure of $A$ coicides with the sequential closure of $A$). Let $(x_n)_{n\in\mathbb N}$ be a sequence in $X$ converging to $x$. ...
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If $x_d \to 0_E$ and $\sup_{d\in D} |\lambda_d| <\infty$, then $\lambda_d x_d \to 0_E$

Let $E$ be a (not necessarily Hausdorff) real TVS. I'm trying to solve exercise 13 in these notes by professor Gabriel Nagy Let $(x_d)_{d\in D}$ be a net in $E$ such that $x_d \to 0_E$. Let $(\...
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A bounded increasing net converges – formulated using filters

We know the following monotone convergence theorem: If $n:X\to\mathbb R$ is a bounded increasing net, then $n$ converges. (Also for other spaces than $\mathbb R$.) I am trying to formulate this ...
Dominique Unruh's user avatar
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Prove that if $\mathcal{F}[x_d]\subseteq \mathcal{G}$, then there's a subnet $(y_e)$ of $(x_d)$ such that $\mathcal{G}=\mathcal{F}[y_e]$.

Let $X$ be a set. Given any net $(x_d)_{d\in D}$ in $X$, we define $$\color{red}{\mathcal{F}[x_d]}:=\big\{F\in 2^X:(\exists d_0\in D)(\forall d\in D)\big(d_0\preceq _Dd\Rightarrow x_d\in F\big)\big\}.$...
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Proving that a sequence $\{\delta_n\} \in \ell_\infty^*$ has no weak-* convergent subsequence but has a weak-* convergent subnet

This is problem 12 in Reed & Simon's book on functional analysis. Let $\{\delta_n\}$ be the sequence in $\ell_\infty^*$ such that $$\delta_n\big(\{c_k\}_{k=1}^\infty\big) = c_n, \quad \forall \{...
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$S$ is compact if every net in $S$ has a convergent subnet

I am going through a proof in Reed & Simon's book on functional analysis that states a space $S$ is compact if every net in $S$ has a convergent subnet. Their proof is: Suppose that every net has ...
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In terms of mappings on topological spaces, are preserving limits and preserving cluster points equivalent?

This question comes from one of my exercises, which asks to prove that a mapping on a topological space is continuous if and only if it preserves cluster points. That is to say, for a topological ...
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Continuity using nets on a dense subspace and regularity

If $Y$ is regular, $f:T\to Y$ is a function such that $X\subseteq T$ is dense in $T$, and for any net $x_\alpha\in X$ converging to $x\in T$, $f(x_\alpha)$ converges to $f(x)$, then $f$ must be ...
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Two nets with a same index set converging to two different points

In a topological space, if we have two topologically distinct points $a,b\in X$, can we always use two nets $(x_i)_{i\in I}$ and $(y_i)_{i\in I}$ with a same index set $I$ to approach each point, ...
user760's user avatar
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Net convergence wrt. intersection of decreasing family of topologies

Let $X$ be a non-empty set and $I=[1, \alpha]$ for some $\alpha>1$. Suppose that $(X, \tau_i)_{i\in I}$ is a family of topological spaces such that $\tau_i\supseteq\tau_j$ whenever $i\leq j$. ...
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Why don't the nets induced by topologies converge to every point?

A net on X is a map w from a directed set to X. In the proofs such as: Let (X, T) be a topological space. If every net in X converges to at most one point. Then the space is Hausdorff. To prove the ...
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Convergence of net in the Hilbert space tensor product of two Hilbert spaces

Let $H$ and $K$ are two Hilbert spaces. Consider the Hilbert space $H \otimes K$, the Hilbert space tensor product of $H$ and $K$. Let $\{ x_l\}_l$ be a net in $H$ and $y$ be an element in $K$. Now ...
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If $Σ$ is endowed with the initial topology $\cal T_ Φ$ corresponding to the collection $Φ$ then $σ_i⟶σ$ iff $φ(σ_i)⟶φ(σ)$ for any $φ∈Φ$.

When I started to study topology of compact convergence it seemed to me that there was something not said explicitly that really help to understand better this topology so that I observed that for any ...
Antonio Maria Di Mauro's user avatar
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Cofinality of a set

Let $(I, \leq)$ be a directed set. Suppose that $I$ has uncountable cofinality and that one can decompose $I$ as a countable union $I = \bigcup_{n = 0}^\infty I_n$, is it true that at least one of the ...
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Net convergence in initial topology

Given a set $X$ and an indexed family $(Y_{i}, \tau_i)_{i \in I}$ of topological spaces with functions $f_{i}: X \rightarrow Y_{i}$. The initial topology $\tau$ on $X$ induced by the collection $(f_i)...
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show if convergent nets get mapped to convergent nets then $f:X \rightarrow Y$ is continuous.

Forgive me if this is a repeat post but it didn't have the solution I was looking for which is one by contradiction. For the following proof, $\mathcal{O}(x)$ is the set of open neighborhoods of $x$. ...
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Finding a net that converges to the zero map in $\mathbb{R}^\mathbb{R}$

I'm looking through Willard's General Topology and I came across this problem in the section on nets. Willard 11A.1 In $\mathbb{R}^{\mathbb{R}}$, let $E = \{f \in \mathbb{R}^{\mathbb{R}} \mid f(x) = ...
Grigor Hakobyan's user avatar
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Definition of upper semi-continuous functions: limsup or liminf?

Notation: $\{f\geq c\}$ stands for $\{x\in x: fx\geq c\}$. The standard definition of an upper semi-continuous function $f:X\to \bar{ \mathbb R}$ is: For each $c$ in $\mathbb R, \{f\geq c\}$ is ...
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$A$ closed and $B$ compact $\implies \exists U$ open convex and balanced with $(A + U) \cap (B + U) = \emptyset$

Let $X$ be a topological vector space, a set $U \subset X$ is said to be balanced if $\lambda u \in U$ for any $u \in U$ and $\lambda \in [-1,1]$ I would like to prove the following theorem Theorem ...
Paul's user avatar
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12 Possible definitions of basis and their properties

Let $I$ be any non-empty set, one can consider the set $2^I_{fin} := \{ J \in 2^I \; : \; J \text{ is finite } \}$ So basically $2^I_{fin}$ consists of all the finite subset of $I$. Then the ordered ...
Paul's user avatar
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Definitions of relative countable compactness

Are the following definitions of relative countable compactness equivalent? $\overline{A}$ is countably compact Every sequence in $A$ has a convergent subnet Note that for regular spaces, the ...
Jakobian's user avatar
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Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

Let $X$ be a topological space. Then I'm trying to prove below result about a function $g$ derived from $f$. Theorem: Let $f:X \to \mathbb R \cup \{\pm \infty\}$. For $a \in X$, let $\mathcal N_a$ be ...
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Let $\alpha := \liminf_{x \to a} f(x)$. Is there a net $(x_t) \subset X$ such that $x_t \to a$ and $f(x_t) \to \alpha$?

Let $X$ be a topological space and $f:X \to \mathbb R$. Let $a \in X$ and $\mathcal N_a$ be the set of all neighborhoods of $a$. Let $$ \alpha := \liminf_{x \to a} f(x) := \sup _{V \in \mathcal N_a} \...
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Compare these definitions of limits of a function giving some references about limits of functions between topological spaces.

Surely in a basic Calculus course it is very common studying limits of functions but surprisingly (at least this is my experience) in Topology it is not common to do this: indeed, between many famous ...
Antonio Maria Di Mauro's user avatar
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Is a topology determined by the convergence of nets?

Consider a space $X$ with two topologies $\tau_1$ and $\tau_2$. It is easy to see that nets remain convergent if the topology is made coarser, i.e. $$\tau_1 \subseteq \tau_2 \quad \Rightarrow \quad \...
MBolin's user avatar
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Prove or disprove if two continuous functions are equal when they agree in a dense set.

So here is proved that two continuous functions $f:X\to Y$ and $g:X\to Y$ from a space $X$ to an hausdorff space $Y$ are equal when they agree in a dense set $D$ of $X$. However it seem to me that the ...
Antonio Maria Di Mauro's user avatar
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Let $A$ and $B$ be nonempty disjoint closed sets such that $A$ is compact. There exists a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$

I'm reading this theorem and its proof. Theorem: Let $A$ and $B$ be two nonempty disjoint closed subsets of a real (not necessarily Hausdorff) t.v.s. $X$ such that $A$ is compact. Then there exists a ...
Akira's user avatar
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Which are more standard: Filterbases or nets? What are the benefits of each?

In the text Topology by James Dugundji there are a lot of impressively short proofs (Ex. Tychonoff. Chap. XI Theorem 1.4(4)) using filterbases that later, when I go to class, take an eternity to prove ...
Choripán Con Pebre's user avatar
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If $X$ is first-countable then a net converges when a subsequence converges?

Let be $X$ and we assume that $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that there exists a cofinal and increasing map $\varphi$ form $\Bbb N$ to $\Lambda$ such that $\big(x_{\varphi(n)}\big)_{n\...
Antonio Maria Di Mauro's user avatar
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How to prove that every Cauchy net in a complete metric space is convergent? [duplicate]

How to prove the basic fact that every Cauchy net in a complete metric space is convergent? My definition: A metric space $X$ is complete iff every Cauchy sequence in $X$ is convergent. I don't ...
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Exercise $11$.D.c from Willard: «Prove that if every subnet of $(x_λ)_{λ\inΛ}$ has a subnet converging to $x$ then $(x_λ)_{λ\inΛ}$ converges to $x$»

In the text General Topology of Stephen Willard is ask to show that if $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that every its subnet has a subset converging to $x$ then $(x_\lambda)_{\lambda\...
Antonio Maria Di Mauro's user avatar
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Let $E$ be a t.v.s. and $f$ a discontinuous linear functional on $E$. There is a net $(x_d)$ such that $x_d \to 0$ and $f(x_d) = 1$ for all $d$

In a previous post, I proved that Let $E$ be a locally convex t.v.s. and $f \not \equiv 0$ a linear functional on $E$. Then $\ker f$ is either closed or dense in $E$. Then I have found a proof ...
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Does every net has a convergent subnet in a compact uniform space?

We know that Result 1: In a metric space compactness implies sequential compactness, that is, if $(X,d)$ is a compact metric space then every sequence $\{x_n\}$ in $X$ has a convergent subsequence. In ...
My Math's user avatar
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Banach limit for nets

It is well known that on the Banach space $l^{\infty}\left(\mathbb{N}\right)$ of bounded functions $f:\mathbb{N}\to\mathbb{C}$ with the sup-norm, there exists a (non-unique) Banach limit. This is a ...
HUO's user avatar
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Countable (Willard) subnet in first-countable space

Let $X$ be a first-countable space, and let $\mathcal B_x=\{B_1,B_2,\dots\}$ be a local base at $x\in X$ satisfying $B_1\supset B_2\supset\dots$. Given a net $(x_\alpha)_{\alpha\in A}$ that converges ...
ho boon suan's user avatar
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Showing that a mapping $f:X \to Y$ is continuous at $z_0 \in X$ when the net $f \circ \varphi$ converges to $f(z_0)$ with $\varphi\to z_0$

Let $(X, \tau_1), (Y, \tau_2)$ be topological spaces, $f:X\to Y$ a mapping, $\varphi:S\to X$ a net converging to $z_0$ and suppose that the composition $f \circ \varphi$ converges to $f(z_0)$. Here $S$...
Cartesian Bear's user avatar
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Let $(x_d), (y_d)$ be nets such that $x_d \to a$ and $y_d \in \overline{\operatorname{conv} \{x_e \mid e \ge d\}}$. Then $y_d \to a$

In solving Ex 3.13.1 in Brezis's book of Functional Analysis. I come across below claims. Let $E$ be a locally convex t.v.s. and $(x_d)_{d\in D}$ a net in $E$ such that $x_d \to a\in E$. Let $$ X_d :=...
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Characterize lower semi-continuous functions by net convergence

In solving this question, I found that s quicker proof is possible with below net characterization of l.s.c. functions. Could you have a check on my attempt? Let $X$ be a topological space and $f:X \...
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