# 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 ...
• 1,514
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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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