# 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 the doubling dimension of any net of a metric space at most half of that of the metric space?

Definition Doubling dimension ($\dim_D(M)=k$): A Metric space $M=(V,d)$ has doubling dimension at most $k$ if for any $x\in V$ & $r>0$, $B(x,r)\subseteq\bigcup^{2^k}_{i=1}B(x_i,r/2)$. With ...
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### How does "inhibitor arc" fit into fundamental equation of Hybrid Petri Nets?

In "ON HYBRID PETRI NETS" by DAVID AND ALLA published in 2001 on page 26 is given an example of how fundamental equation solves a HPN for given start and end time values. A system looks like And ...
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### A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
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### What should the morphisms in the Category of Directed Sets be?

Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...
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### Connection between subnet and superfilter

Let's define a net and subnet in this way: A net is any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set. A net $m:(P',\le)\to X$ is a subnet of the net \$n:(P,\le)\...
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### Direct Limits and Limits of Nets

A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...
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