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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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 ...
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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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Does this convergence characterization hold for nets in a metric space?

Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$, $$ x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
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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) = ...
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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 ...
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An alternative proof of Ascoli-Arzela Theorem

Let $(X,d)$ be a compact metric space, let $C^{0}(X,\mathbb{R})$ be the vector space of continuos functions from $X$ to $\mathbb{R}$ equipped with the sup norm $||f|| := \max_{x \in X}{|f(x)|}$ Let $(...
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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 ...
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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 ...
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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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$\liminf$ is the smallest cluster point in the setting of net convergence

I'm trying to prove that $\liminf$ is the smallest cluster point in the setting of net convergence. Could you have a check on my attempt? Let $X$ be a topological space and $f:X \to \mathbb R$. Let $...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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Let $B$ be a filterbase subordinate to $A$. Then, if $\varphi$, $\phi$ are nets that determine $A$, $B$, then $\varphi$ is subnet of $\phi$.

I want to show that the notion "subordinate" is equivalent to that of subnets. That is Let $Y$ be a set. Let $\varphi\colon D\to Y$ and $\phi\colon \Delta\to Y$ be nets and $\mathfrak{B}$, $...
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When a transfinite sequence converges with respect the order topology? Is the limit of a transfinite sequence necessarly its supremum?

Let be $s_\theta$ a transfinite sequence $(\alpha_\nu)_{\nu\in\theta}$ of length an ordinal theta $\theta$. So any ordinal is totally ordered with respect the usual order relation so that any ordinal ...
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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\...
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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\...
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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 ...
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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 ...
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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 ...
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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 $S$...
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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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Lower semi-continuous function attains minimum on a compact set [duplicate]

I'm trying to use net to prove this well-known result. Could you have a check on my attempt? Let $E$ be a compact topological space and $f:E \to \mathbb R$ lower semi-continuous. Then $f$ attains the ...
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Proof of Hausdorffness of sequentially Hausdorff space under its sequential topology

Under "Topology of sequentially open sets" section of the Wikipedia page Sequential Space, there is a claim which says any sequentially Hausdorff(i.e. every convergent sequence has a unique ...
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Is there a common directed set that induces $2$ monotone cofinal maps to $2$ other directed sets?

Recently, I have come across a result that the diameter of set is equal to that of its weak closure. A proof is straightforward if below result is true. Let $E$ be a locally convex Hausdorff t.v.s., $...
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If a weakly convergent net is bounded by $r$, is its limit bounded by $r$?

Let $(E, | \cdot |)$ be a normed space and $E'$ its topological dual. Let $\sigma(E,E')$ be the weak topology of $E$. Let $(x_d)_{d\in D}$ be a net in $E$ that converges in $\sigma(E,E')$ to $x\in E$. ...
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How can I work with the net associated to a filter?

I have a question about the net which one can always associate to a filter. First let me write down our definition: If $\mathfrak{F}$ is a filter on $M$ then we define $$I_\mathfrak{F}=\{(A,p): A\in \...
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A proof of Banach–Alaoglu theorem with net convergence

I'm reformulating the proof of Banach–Alaoglu theorem in Brezis's book of Functional Analysis. My goal is make his argument clearer. I'm very happy to use net convergence to characterize compactness ...
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How do I prove the following statement about filters and associated nets?

I have the following problem: Let $\mathfrak{F}$ be a filter with associated net $(p_i)_{i\in I_\mathfrak{F}}$. Show that $p\in M$ is a cluster point of $\mathfrak{F}$ iff $p$ is a cluster point of ...
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Is the intersection of a decreasing (w.r.t. inclusion) sequence of cofinal subsets again a cofinal subset?

Let $(D, \le )$ be a directed set. A subset $A$ of $D$ is cofinal in $D$ if and only if $\forall d \in D, \exists a\in A, d \le a$. Let $(D_n)$ be a decreasing (w.r.t. inclusion) sequence of cofinal ...
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A simpler equivalent definition of subnet

I'm reading the definition of subnet from here, i.e., Let $E$ be a non-empty set and $(x_d)_{d\in D}$ a net in $E$. Let $(y_t)_{t\in T}$ be another net in $E$. Then $(y_t)_{t\in T}$ is called a ...
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Entourage asymmetry and Cauchy nets: $x_a$ is $V$-close to $x_b$ does not imply $x_b$ is $V$-close to $x_a$

I have a few quibbles about the nature of the ordering $(a,b)$ versus $(b,a)$ when it comes to membership of an entourage in a uniform space. The Wikipedia article on uniform spaces nowhere asserts ...
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Conjugate Nets in Higher Dimensions

Consider the following definition of conjugate nets in arbitrary dimensions (in the following $3 \leq N$ and $2\leq m < N$). Definition: An immersion $f: \mathbb{R}^m \longrightarrow \mathbb{R}^N$ ...
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Question on states and nets in a non-unital $C^*$ algebra

Let $A$ be a non-unital $C^*$ algebra, and let $a_\lambda$ be a net such that $a_\lambda b \to 0$ for all $b \in A$. Show that for any state $f, f(a_\lambda) \to 0$. I tried using approximate units, ...
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A direct proof (using net convergence) that sequential compact metric space is compact

Let $X$ be a metric space and $A \subseteq X$. If there is a net $(x_d)_{d\in D}$ in $A$ that converges to $a \in X$, then there is a sequence $(y_n)_{n\in \mathbb N}$ in $A$ that converges to $a$. So ...
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$x_d \to a$ in $\tau$ if and only if $x_d \to a$ in $\tau_A$

I'm trying to prove below equivalence. Could you verify if my understanding is correct? Let $(X, \tau)$ be a metric space, $A \subseteq X$, and $\tau_A$ the subspace topology of $A$. Let $a\in A$ and ...
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$3$ equivalent definitions of compactness

I'm trying to prove this equivalence. Could you have a check on my attempt? Let $X$ be a topological space. The following statements are equivalent. (S1) Every open cover of $X$ has a finite ...
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Equivalent definitions of cluster point of a net

In proving the equivalent definitions of compact sets, I come across below equivalence of cluster point. Could you have a check on my proof? Let $X$ be a topological space, $(x_d)_{d \in D}$ is a net ...
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$f$ is continuous if and only if, $f(x_d) \to f(x)$ where $(x_d)_{d \in D}$ is a net such that $x_d \to x$

I'm trying to show this equivalence of different formulation of continuity. Could you have a check on my proof? Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces and $f:X \to Y$ be continuous,...
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$\overline A$ is the set of limits of convergent nets with values in $A$

In proving that the equivalence between different definitions of continuous functions between topological spaces, I come across this lemma. Could you have a check if my attempt is correct? Lemma: Let ...
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How can I think about a net in topology? [duplicate]

I want to ask if someone could give me some intuition about nets in topology. I know the following definition: Let $(M,\tau)$ be a topological space an $(I,\leq)$ a directed set. This means that $I$ ...
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