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 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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A net converges to a point iff every subnet accumulates in that point. [duplicate]

While working on a takehome for my functional analysis course I stumbled upon this small lemma A net $(x_i)_{i\in I}$ in a topological space $X$ converges to a point $x\in X$ if and only if every ...
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Extracting countably many subnets from a net

Let $\{f_{\alpha,k}\}_{\alpha\in \mathcal A,k\in \mathbb N}\subseteq \mathbb R$, were $\mathcal A $ is a directed set. Suppose I know that for each $k$, and $\varepsilon >0$, there is $\bar \alpha ...
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Reference for convergent nets as continuous functions from $D\cup\{\infty\}$

If we have any directed set $(D,\le)$ then we can add a point $\infty\notin D$ and then consider the topology on $C(D)=D\cup\{\infty\}$ such that all points of $D$ are isolated and the local base at $\...
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Prove a map $\varphi$ between all nets and all filters is surjective.

We define such a map: $\varphi(N) = \{A \subset X \ | \ \exists \alpha \in \Omega, \forall \beta \ge \alpha: f(\beta) \in A \}$ Here $N = (\Omega, f, \ge)$ is a net with index set $\Omega$. This way $\...
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How to explain "mathematically" why not every net is a sequence?

A past question ask to "Give an example of a net that is not a seqence and explain why mathematically". I know that the function $f:((0,1),\leq)\rightarrow\mathbb R$ is a net (since $((0,1),\...
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A increasing function between two directed sets doesn't guarantee a subnet

The following is a definition of subnet using a function between two directed sets: (cropped scan) Def. Let $P: \Lambda \rightarrow X$ and $Q: M \rightarrow X$ be nets (where $\Lambda$ and $M$ are ...
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For non continuous functions there exists a net for which the "image-net" does not converge (Exercise)

I have a Question about the following exercise: Let $X,Y$ be topological spaces and $f:X \rightarrow Y$ in $x_0$ not continuous. Proof that there exists a net $(x_i)_{i \in I}$ in $X$ such that for $...
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Is the derivative of a complex function uniquely determined?

Definition A complex function $f:\Omega\rightarrow\Bbb C$ defined in a open set $\Omega$ of $\Bbb C$ is said derivable at $z_0$ if the limit $$ \frac{df}{dz}(z_0):=\lim_{h\rightarrow0}\frac{f(z_0+h)-f(...
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Do convergent nets have discrete subnets?

Let $X$ be a Hausdorff space. Assume that $x'\in X$ and $(x_n)$ is a sequence convergent to $x'$ and such that $x_n\neq x'$ for any $n$. Then $\{x_n\}$ as a subspace of $X$ has to be discrete because ...
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A different definition of convergence of series

I am reading the introduction to Hilbert space by P. R. Halmos. I found there the following definition of convergence of a series: A family $\{x_j\}$ of vector will be called summable with sum $x$, in ...
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Generalizing sequences

This is a follow-up on my previous question: Generalizing the idea of sequences Suppose $S$ is a non-empty set with no metric defined on it. Which of the following definitions could be extended so as ...
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Routine verification of net convergence

Question $1$: Suppose that $(D,\le '), (E, \le '')$ are two directed sets where $D \cap E = \varnothing$. We impose an order $\le$ on $D \cup E$ thus: $x \le y \iff \cases{x, y \in D, x \le′ y \text{...
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Exercise about convergence of a net in $(\mathbb R, T_{CF})$

Let $(\mathbb R, T_{CF})$ topological space with $T_{CF}=\{U\subset \mathbb R\mid \mathbb R\setminus U \text{ is finite}\}$ the co-finite topology. Let $s=(s_d)_{d\in D}$ a net in $\mathbb R$ such ...
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$K\subseteq X$ is compact iff every net in $K$ has an accumulation point in $K$

Is it possible to prove the statement Let $X$ be a topological space. Then $K\subseteq X$ is compact iff every net in $K$ has an accumulation point in $K$. based on the following theorem, which I ...
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If $\sum_{s \in S} \|h_s\| < \infty$ then $\sum_{s \in S} h_s$ converges.

Let $H$ be a Hilbert space and $\{h_s: s \in S\}$ be a set of vectors in $H$. Assume that $\sum_{s \in S}\|h_s\| < \infty$. Can we conclude that $\sum_{s \in S} h_s$ converges in $H$ in norm? Here, ...
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Limit set of a net in $(\mathbb R, T_{CF})$

Let $(\mathbb R, T_{CF})$. Let $s=(s_d)_{d\in D}$ a net in $\mathbb R$. Proof that: $$\mathcal Lim(s) =\mathbb R\iff \bigcap_{d\in D}B_d=\emptyset$$ Notation $T_{CF}=\{\emptyset, \mathbb R\}\cup \{A\...
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On Tichonoff's theorem and convergent subnets

The space of functions $[0,1]^{[0,1]}$ is compact in the topology of pointwise convergence due to Tichonoff's theorem. Consider the sequence of functions $\{f_n\}_n$ defined by $$f_n(x)=\left\{\begin{...
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Convergence of product of two convergent nets in a topological group.

Let $G$ be a topological group and $(x_j)_{j \in J}$ and $(y_j)_{j \in J}$ be two nets in $G$ indexed by the same directed set $J$ converging to $x$ and $y$ in $G$ respectively. Can it be said that $...
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Convergence of net in $C^\infty(0,\infty)$ implies pointwise convergence.

Consider the space $C^\infty(0,\infty)$ of functions $f: (0,\infty) \rightarrow \mathbb{R}$ which have continuous derivatives of all orders. Equip $C^\infty(0,\infty)$ with the topology induced by ...
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Continuous function and net (Solution Verification)

I have started to learning about nets. I need a solution verification or any necessary comments (answer if needed) for the following problem: Statement: Let $X$ be a topological space and let $f:X\to \...
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can one speak of $\max(i,j)$ for elements $i,j$ of a directed set?

So i made a convergence argument using nets where i used the triangle inequality. For a simplified example of what im talking about lets say i have something like a net $(a)_{i\in I}$ in $\mathbb{R}$ ...
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sequence with an accumaltion point, but no converging subsequence

I try to think of an example of a sequence (in a topological space) that has an accumulation point but no subsequence converging to it. I found this question to this theme Example of converging ...
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Non-dense subspace implies non-zero measure

I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part. Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the ...
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To prove: if X' is not compact, then there is a countably infinite discrete set in X.

In a research article I found: In a metrizable topological space $(X,\tau)$, if set of limit points $X'$ is not compact, then $X'$ contains a countably infinite discrete set. I don't understand how ...
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Is there a basis for sequences?

There are various equivalent definitions of a topological space and for some of them we have the concept of basis: a basis of opens sets, or a basis of neighbourhood. This concept simplifies verifying ...
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Weak vs. Weak* Convergence of Bounded Nets

Let $X,Y$ be Banach spaces. In V. Paulsen's book on completely positive maps, he shows that $B(X,Y^\ast)$ is a dual space as follows: For $x \in X, y \in Y$ let $x \otimes y \in B(X, Y^\ast)^\ast$ be ...
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Example of an unbounded convergent net in a metric space

Please could someone explain how we can obtain a convergent net in a metric space that is unbounded. I have considered indexing a subset in $\mathbb{R}$ by numbers in $(0,\infty)$ as I thought having ...
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How to define a topology with a given convergence of nets.

Suppose $X$ is a non-empty set and $\{\tau_{i}\}_{i}$ be a family of topologies on X. I want to define a topology $\tau$ on $X$ so that: A net $\{x_\alpha\}_\alpha$ converges to $x$ w.r.to $\tau$ if ...
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Negation statement a net

Let $X$ be a topological space and $A\subseteq X$. Then $A$ is closed if and only if every net in $A$ that converges to some $x\in X$, we have $x\in A$. How do I negate the statement "every net ...
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Nets: frequently in $A$ iff eventually not in $X\setminus A$?

I'm reading these notes on the proof of Tychonoff's theorem using nets. I'd appreciate a proof verification, and an alternative proof strategy for the second lemma. Directed set: A directed set is a ...
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Topology axioms in terms of net convergence

I am looking for the list of axioms of "net convergence" in the language of nets which correspond to the axioms of a topology. (Notice that neither Wikipedia nor nlab seem to answer this ...
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Ex.11 Sec.I-4 Conway's Functional Analysis [closed]

The following is from chapter 1, section 4 of Conway's A Course in Functional Analysis, second edition: I tried a lot but have no idea how to solve it : If ${\{h_n}\}$ is a sequence in a Hilbert ...
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Does the concept of a positive null net make sense for all directed sets?

A common strategy to prove things is to use null sequences and I was wondering if it is possible to generalise the concept to aritrary nets. Let $I$ be a directed set. Is there a net $\left( x_\alpha \...
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Construction of a net and the usage of Axiom of Choice

I read in a homepage about nets, where the author gives an example (minor modification for context) Example. Given a topological space $X$ and a point $x\in X$, let $N_x$ denote the directed set of ...
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How does one show that if $B \subset A$ (where $A$ is a directed set) is cofinal in $(A, \leq)$ then $(B, \leq)$ is a directed set?

First two axioms of directed sets readily follow for $(B,\leq)$ by the virtue of being a subset of $(A,\leq)$ but I don't see how the third one follows. Directed set $(A,\leq)$ is a set with the order ...
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Let $(a_d)_{d\in D}$ be a real net such that $\lim_{d\in D}a_d=∞$. Is there a cofinal set $D'\subset D$ such that $(a_d)_{d\in D'}$ is increasing?

Let $(D, \geq)$ be a directed set, and let $(a_d)_{d\in D}$ be a real-valued net satisfying $$ \lim_{d\in D}a_d =+\infty. $$ Can we find a cofinal subset $D'\subset D$ such that the restricted net $(...
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Topology generated by those subsets of $X$ which are preserved by images of nets in that subset by a function [duplicate]

$\mathbf {The \ Problem \ is}:$ Let, $\mathcal N(X)$ be the set of all nets in a topological space $X$, and $\mathcal P(X)$ be the power set of $X.$ Let, $L : \mathcal N(X) \to \mathcal P(X)$ be a ...
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Is intersection of a subset with a net a subnet?

Say $\{x_i\}_{i \in I}$ is a net in a set $X$ and $Y\subset X$ then, is $\{x_i\}_{i \in I}\cap Y$ a subnet? intuitively this seems true but I don't know how to approach the proof.
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$N(X)$ denotes collection of all nets of $X$. $T:N(X)\to \mathcal{P}(X)$ be a map such that $T\{x_\delta\}=T\{y_\gamma\}$ if one is subnet of other.

Let $X$ be a non-empty set. $N(X)$ denotes the collection of all nets on $X$ and $\mathcal{P}(X)$ denotes the collection of all subsets of $X$. Let $T:N(X)\to \mathcal{P}(X)$ be a map satisfying the ...
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Question about terminology regarding nets

Show that $U$ is an open subset of a topological space $X$ if and only if for every point $z\in U$ each net converging to $z$ is co-finitely contained in $U$. Is this question as stated correct? IMO, ...
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Proving uncountable sum as series using nets.

I saw this question: The sum of an uncountable number of positive numbers Asking about a proof of the following: Let $A = \{a_i\}_{i\in I}$ be a set of positive numbers. If the uncountable sum ...
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Assume a real-valued net has a cluster point $a\in \mathbb R$. Is there a cofinal subnet converging to $a$?

Let $(D, \geq)$ be a directed set and let $(n_d)_{d\in D}$ be a real-valued net. Assume $a\in \mathbb R$ is a cluster point of $(n_d)$, i.e., for every neighborhood $U$ of $a$ and every $d\in D$ there ...
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'Cofinality' and base cardinality of convergent net less than cardinality of index set, implies constant from some index onwards?

I have seen the following statement: Let $\{ x_i\}_{i\in I}\subseteq \mathbb{R} $ be a net converging to an element $x_0\in \mathbb{R}$, where $\vert I\vert>\aleph_0$, $I$ is totally ordered and ...

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