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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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A question about continuous actions of topological groups.

Let $G$ act continuously on $X$. If $K$ is a compact subspace of $G$ and $B$ a closed subspace of $X$, is $KB:=\{k.b|k\in K,b\in B\}$ a closed subspace of $X$? I know this to be true when $X$ is the ...
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Prove any subnet of a convergent net is convergent

Let $f:J \rightarrow X$ be a net in $X$, let $f(\alpha) = x_\alpha$. If $K$ is a directed set and $g:K \rightarrow J$ is a function such that $i \leq j \rightarrow g(i) \leq g(j)$ and $g(K)$ is ...
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Can we define the unordered sum of a set of hyperreal numbers?

If $\{a_i:i\in I\}$ is a set of nonnegative real numbers, then the unordered sum $\sum_{i\in I}a_i$ is defined as $\sup \Bigl\{ \sum_{i\in A}a_i\,\big| A \text{ finite, } A \subset I\Bigr\}$. And $\...
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Standard definition of subnets.

Reading Willard's General Topology, I found the following definition of a subnet of a net $f:D\to X$: $g:E\to X$ is a subnet of $f$ if there exists an increasing $\varphi:E\to D$ such that $g=f\...
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37 views

Diagonal-type argument for real nets

Let $X \subseteq \mathbb{R}$, and suppose we have the following property: For every $\varepsilon > 0$ there exists a sequence $(x_n)_{n = 1}^{\infty} \subseteq X$ such that $$ \limsup_{n \to \...
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Must a Convergent Net in a Normed Space be Bounded?

If $ X $ is a normed space and $ (x_n)_{n=1}^{\infty} \subset X $ is a convergent sequence, then it is elementary to show that $ \| x_n \| $ is bounded by observing that there exists an $ N \in \...
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Open set in terms of nets

Let $X$ be a topological space, and $U\subseteq X$ be a subset of $X$ with the following property: For every convergent net $x_\alpha\to x$ in $X$ such that $x\in U$, there exists an $\alpha$ such ...
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Two Hausdorff topologies on the same set with the same convergent nets

Let $X$ be a set and $\mathcal{T_1,T_2}$ two Hausdorff topologies on $X$ such that they admit the same convergent nets, i.e., a net $(x_{\alpha})_{\alpha}$ in $X$ converges with respect to $\mathcal{...
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44 views

Convergence of monotone nets

In sequences of real numbers, we have a monotone convergence result: If $a_{n+1}\geq a_n$ and bounded, then $a_n$ converges to it's supremum. The proof seems to work also in the net case. My ...
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Final maps with the same domain

Let $\mathbb D$ and $\mathbb E$ be two directed sets, then a map $f:\mathbb D\to \mathbb E$ is said to be final,if for any $e\in E$ there exists some $d\in D$ such that $f(d')\geq e$ whenever $d'\geq ...
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Define non-eventually-constant $f: I \to \{a, b\}$ from arbitrary upwards-directed poset $I$

Is the following provable and how? I feel like I am missing some proof technique or strong theorems, I'd be grateful for any pointer. Let $(I, \leq)$ be an upwards-directed poset. Define an $f: I \...
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A net has a limit if and only if all of its subnets have limits (without the use of Cauchy nets)

I am trying to prove the above. More specifically I am trying to prove the direction "if all subnets of a given net have limits then the net in question has a limit" The definition I am using for a ...
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33 views

Any Cauchy net in metric space is convergent?

Any Cauchy net in a metric space is convergent? I know that if (X,d) is metric space COMPLETE then cauchy net in X is convergent in X. But, only (X,d) is metric space (not complete) It is true? pd: ...
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Problem in proof of -“A net has $y$ as a cluster point iff it has a subnet which converges to $y$”

Directed Set: We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$ such that (i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$; (ii) x ≤ x for each x ∈ $\omega$; (iii) ...
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If $A\subset Y$,then $f^{-1}(A)=X-f^{-1}(Y-A)?$

If $(x_{\lambda})$ is an ultranet in X and $f:X\rightarrow Y$ then $(f(x_{\lambda}))$ is an ultranet in Y. Proof: If $A\subset Y$,then $f^{-1}(A)=X-f^{-1}(Y-A),$so ($x_{\lambda}$) is eventually ...
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Is convergence of a net of functions almost everywhere a topological notion?

There exists no topology on the set of Lebesgue-measurable functions such that a sequence is convergent in that topology if and only if it is convergent almost everywhere (i.e. everywhere except for a ...
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What kinds of nets can be used to singlehandedly guarantee continuity?

Let $X$ and $Y$ be topological spaces. Then for any directed set $A$ we can define nets in $X$ and $Y$ indexed by $A$. And a function $f:X\rightarrow Y$ is continuous at a point $x$ if and only if ...
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How to show there is no point in boundary for a convergent net in a closure?

Let $X$ be a topological space. Let $M\subseteq X$. Let $(x_i)_{i\in I}$ be a net in $\overline{M}$ such that $x_i\rightarrow x\in\overline{M}$ and $x_i\neq x$ for all $i$. Then how to show $x_i\in M$ ...
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A Corollary from Munkres' Topology

This exercise comes from Munkres' Topology, 2nd edition, Page 188. It says Corollary. Let $G$ be a topological group; let $A$ and $B$ be subsets of $G$. If $A$ is closed in $G$ and $B$ is compact, ...
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Closure operator via nets, regarding the fact that $\overline{A \cup B} = \overline{A} \cup \overline{B}$

Let $(X, \tau)$ be a topological space, and let $\overline{ (-) } : \mathcal{P}(X) \to \mathcal{P}(X)$ be defined as $$ \overline{S} := \{x : \exists(x_\alpha)_{\alpha \in \Lambda} \subseteq S \text{ ...
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Find really simple error in “proof” every subnet has a limit

I was trying to use the language of nets and subnets. But I noticed I was essentially proving every subsequence $x_1, x_2, \ldots $ has a convergent subsequence - namely $x_1, x_1, x_1, \ldots$ - ...
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A net vs a sequence

A net is defined as a map $\Theta\to \mathbb{X}$ ($\theta\mapsto x_{\theta}$) where $\Theta$ is a directed set and $\mathbb{X}$ is some topological space. If $\Theta=\mathbb{N}$ then this definition ...
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Is a converging net + limit always compact?

Let $X$ be a topological space. It is known that if $\{x_n\}_{n \in \mathbb{N}}$ is a sequence in $X$ that converges to $x$, then the set of points $\{x,\{x_n\}_{n \in \mathbb{N}}\}$ is compact. Is ...
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Double Limit of operators converges weakly, does single limit converge?

I am working on a problem from Reed and Simon, which states: Suppose $\{A_\alpha\}$ and $\{B_\alpha\}$, $\alpha \in I$, are nets. Let $A_\alpha^* \to A^*$ and $B_\alpha \to B$ in the Strong Operator ...
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Showing that a net is universal

Let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$ and let $x:\mathcal{U} \rightarrow \mathbb{N}$ be a net such that $x(U) \in U, \forall U \in \mathcal{U}$. Show that $x$ is universal. I tried some ...
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Cauchy nets in products of uniform spaces and their projections

I am stuck trying to prove why a net in a product of uniform spaces is Cauchy if and only if every projection of it is a Cauchy net. I assume, analogously to the fact that continuous uniformity ...
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Convergent Nets and Composite Functions

Does anyone have any idea how to prove the following...? Let $X, Y$ be topological spaces and let $g:X\rightarrow Y$ be a map. If for every directed set $I$ and convergent net $f:I\rightarrow X$ with ...
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Compactness implies that every net has a converging subnet - why that definition of subnet?

The definition of subnet is more convoluted than expected. The idea seems to be that the definition is such that the equivalence compactness $\Leftrightarrow$ every net has a convergent subnet holds. ...
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Does convergence in net imply convergence in sequence?

Let $(X,\tau)$ be a topological space, let $x^*$ be an element of $X$, and let $(x_{\alpha})$ be a net from some directed set $A$ into $X$, that converges toward $x^*$. Is there necessarily some ...
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net filter in topology [closed]

Let $X$ be a topological space with a subbase $\mathcal{S}$ and $x$ belongs to $X$. Prove that a net in $X$ converges to $x$ iff the condition in the definition of convergence of nets holds for all ...
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Cardinal of the domain of a net

Definition: A net $\nu$ is said to be a subnet of net $\nu',$ if there exists some function $\varphi:\mathrm{dom}(\nu)\to \mathrm{dom}(\nu'),$ such that $\bf 1)$ $\nu=\nu'\circ\varphi,$ $\...
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The topology induced by a “good” net convergence notion induces a net convergence notion as originally specified

The following is from Problem 11D in Willard's General topology textbook. Suppose we have some notion of convergence on a set $X$ satisfying the following properties. Fix $x\in X$ and let $I$ be a ...
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When does order convergence imply topological convergence?

Let $V$ be a locally convex topological vector space (or even a Banach space), equipped with a closed partial order, or equivalently, a closed positive cone. It is well known that any directed net $\{...
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Does every net have a countable subnet?

I was wondering whether a net always has a countable subnet? Since there is a criterion for continuity using nets, and in some spaces we can check only for sequences. It would seem to me that a ...
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Double limit of a net

Assume that $(m_{\alpha})_{\alpha \in \Lambda}$ is a bounded net in Banach algebra $A$ and $(n_{\gamma})_{\gamma\in \Gamma}$ is a bounded net in Banach algebra $B$. Let $(t_{(\alpha,\gamma)})_{(\...
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Net convergence and neighborhood of the limit point

Let $x \in X$. If for every net $(x_{\lambda})_{\lambda \in \Lambda}$ in $X$ such that $x_{\lambda} \rightarrow x$ there exists $\lambda_0 \in \Lambda$ such that $x_{\lambda} \in A \ \ \ \forall \...
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Infimum and supremum And Net

We know that every Partially Ordered Set has to satisfy three conditions : Reflexive Anti-Symmetric Transitive If we have the partially ordered set $S$ with a relation $R$, and $S$ also satisfies ...
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58 views

Existence of a monotone (and cofinal) choice function

Suppose $A$ and $B$ are directed sets, and $\{A_b\}_{b \in B}$ is a collection of residual subsets of A indexed by B. Does there necessarily exist a monotone choice function $f: B \to A$ such that $f(...
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eventually and frequently for nets

I'm studying nets and there is something in the definition of eventually and frequently that is confusing me. These are the definitions I have. Eventually: a net $(x_\lambda)_{\lambda \in \Lambda}$ ...
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How to characterize the net convergence in final topology?

Given a set $X$ and an indexed family $\{(Y_i,\mathscr T_i)\}_{i\in I}$ of topological spaces with functions $f_i:Y_i\to X$. Let $\tau_{\text{final}}$ be the final topology in $X$ w.r.t. $\{f_i\}_{i\...
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Show that a particular subnet is not a subsequence

Here, $\mathbb{N}$ designates the unordered and undirected set of nonnegative integers. Let $A=\mathbb{N}$ be directed by $\leq$ and let $B=\mathbb{N}$ be directed by the relation $\preccurlyeq$ that ...
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Subnet of a sequence.

Denote by $I$ the interval $[0,1]$. The space $\{0,1\}^I$ can be viewed as the space of all functions $$f: I \to \{0,1\}.$$ With the product topology (or equivalently, the point to point convergence ...
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$\{x_j\}_{j\ge i}$ is a subnet of $\{x_i\}_{i}$?

Let $\{x_i\}$ be a net. Then, for any $i$, we take $\{x_j\}_{j\ge i}$. Then, $\{x_j\}_{j\ge i}$ is a subnet of $\{x_j\}_{i}$. Is the above statement correct? I have checked wikipedia for many times....
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Topology, equivalent statements, nets, boundary

Let $(X,\tau)$ be a topological space and $V\subseteq X$. Then is equivalent: 1) $x\in\partial V$ 2) For every neighborhood $U$ of $x$ is $V\cap U\neq\emptyset\neq (X\setminus V)\cap U$ ...
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Topological space, equivalent statements, net, closure, neighborhood

Let $(X,\tau)$ be a topological space and $V\subseteq X$. For $x\in X$ are equivalent: I) $x\in\overline{V}$ II) For every neighborhood $U$ of $x$ is $U\cap V\neq\emptyset$. III) It exists a net $(...
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What are some examples in which the introduction of nets helped understand a concrete topological space?

I've had the chance to learn about nets, though every statement I was exposed to didn't seem to be useful in practice. For example, the fact $x \in \bar{A}$ iff there exists some net $(x_\alpha)_{\...
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Convergent net which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...
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1answer
218 views

A net converges to a point iff every subsequence of the net converges to the same point in first countable topological spaces

I'm having trouble proving the fact that in first countable topological spaces, a net converges to a point iff every subsequence of the net converges to the same point. I first encountered this ...
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1answer
127 views

If every convergent net in a space is eventually constant then the space is Discrete.

The question I am asking is actually I have asked here. I have solved the problem in the following way: Proof: We have every convergent net in $X$ to be eventually constant....$(1)$ We have to ...
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225 views

Characterization of a topological space in terms of convergence of nets in it.

Can we characterize a discrete topological space $X_\mathcal d $ by the convergence of nets in it, if yes then how ? l know that any convergent net in $X_d$ is eventually constant, but does the ...