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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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 ...
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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 ...
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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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Existence of universal subnets via lattice theory

I refer systematically to J.L. Kelley's book on General Topology, in particular pages 80-81. In exercise 2.I, the following theorem is stated: Theorem. Let $A$ and $B$ be disjoint subsets of a ...
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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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Are upward directed set equivalent to downward directed set ? Example with $\supseteq$ and its dual relation $\subseteq$

If the set of partitions of $[a,b]$ $S$ is taken as the directed set, along with the order $\subseteq$, then the directed set has the property that "for every pair $(c,d)$ in the directed set ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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