# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...