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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Why is sequences enough in the definition of a normal family
In the definition of a normal family in complex analysis, we are concerned with a sequence of functions having a subsequence uniformly converging to a function on compact subsets of an open domain. ...
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If $(y_\lambda)_{\lambda \in \Lambda}$ is a subnet of $(x_\alpha)_{\alpha \in A}$ and if $A$ is a complete lattice, is $\Lambda$ a complete lattice?
Let $X$ be a set, $(x_\alpha)_{\alpha \in A}$ be a net in $X$ and $(y_\lambda)_{\lambda \in \Lambda}$ be a subnet of $(x_\alpha)_{\alpha \in A}$.
If $A$ is a complete lattice (i.e. not only a directed ...
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How to construct the nets using the coordinate projection on the cube?
I am reading a paper, but I don't quite understand how they construct the nets for a set. I hope someone can explain why this method works. I have previously encountered how to construct nets on a ...
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how to understand the epsilon-nets are constructed in the followsing set about the $n$-dimentisonal vector?
I am reading an article about constructing nets on a set, but I do not fully understand how the epsilon-nets are constructed. The general idea is to partition the size of the coordinates of a vector ...
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Non-overlapping subnets always exist?
Suppose a net $(\alpha_i)_{i\in I}$, where $I$ is a directed set with no maximal element and $X$ a topological space, has no duplicate values. That is, for $i\neq j\in I$, $\alpha_i\neq \alpha_j$. Is ...
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Convergence of a net and its associated filter
I am reading "Topology: An Introduction" by Waldmann and I am trying to prove the result (iv) in Proposition 4.2.6:
A net converges to $p$ $\iff$ its associated filter converges to $p$.
I ...
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How are $\epsilon$-nets (as in centers of $\epsilon$-ball covers) related to nets (as in topology)?
Here are two definitions that I have encountered:
The first, corresponding to this Wikipedia page, is the following.
Definition.$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$....
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Testing points on the closure of a union of increasing subsets
Let $X$ be a topological space. Let $(S_\lambda)_{\lambda\in\Lambda}$ be a net of closed subspaces of $X$, and suppose that for all $\lambda\le\mu$, $S_\lambda\subseteq S_\mu$.
Let now
$$
x\in \mathrm{...
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How to formulate the condition $\forall t \in [0,1]: p_t(x_n-x) \underset{n \rightarrow \infty}{\rightarrow} 0 $ in the sense of net convergence
Consider the following:
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real valued functions $[0,1] \rightarrow \mathbb{R}$. And $\mathcal{P}:={p_t:t \in [0,1]}$ a family of seminorms such that $p_t(...
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Filtration of separable Hilbert spaces
Let $\mathcal{H}$ be a separable Hilbert space. Let $(\mathcal{S}_\alpha)_{\alpha\in\Lambda}$ be a decreasing net of closed subspaces of $\mathcal{H}$, i.e. such that for each $\alpha,\beta\in\Lambda$,...
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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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