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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155 views

Relative compactness in topological spaces (reference request)

Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis: 1a) The set $S$ is compact if and only if each sequence in $S$...
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Do multiplicative Banach limits exist?

Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies $$\sup_{d \in D} \inf_{c \...
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73 views

Lower semicontinuity of a multi-valued map $F:X\to 2^Y$ in term of net

Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is lower semicontinuous at $x_0\in X$ if for each $y_0\in F(x_0)$ and any neighborhood $U\in \mathcal N(...
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0answers
60 views

Quantification over Nets

On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$. With this, compactness of $X$ (for instance) is equivalent to "every net $(...
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Reference Request: Egoroff Theorem for nets

Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?
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1answer
175 views

Convergent net in a quasi-uniform space 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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1answer
195 views

Characterization of nets with no convergent subnets in Banach spaces

Let $X$ be a finite-dimensional Banach space and $(x_i)_{i\in I}$ a net in $X$. Since every limited net in $X$ has a convergent subnet, it follows that $(x_i)_{i\in I}$ does not admits a convergent ...
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1answer
571 views

The need for nets in topology

I remember when I first heard about nets in topology (called also Moore-Smith sequences). I was told that most of useful topological properties which can be exressed in terms of sequences in the ...
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2answers
1k views

What is a generalized limit?

In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any ...
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1answer
120 views

How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?

Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function $...
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1answer
93 views

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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1answer
246 views

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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1answer
193 views

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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1answer
474 views

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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1answer
398 views

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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432 views

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 ...