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.
21
questions
12
votes
11
answers
4k
views
Are nets and filters useful in geometry and topology?
Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...
- 15.1k
8
votes
1
answer
620
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 $p \preceq r$ ...
- 352
6
votes
1
answer
808
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 ...
- 9,150
6
votes
1
answer
651
views
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 \...
- 839
5
votes
1
answer
439
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)\...
user31967
4
votes
2
answers
2k
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 ...
user1688
4
votes
1
answer
254
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 ...
- 195
3
votes
2
answers
340
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 $S$...
- 11.6k
2
votes
2
answers
529
views
A net of lower semicontinuous functions
Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise.
Please is it true that one can extract a countable ...
2
votes
1
answer
207
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 ...
- 1,220
2
votes
1
answer
274
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 ...
- 59
2
votes
1
answer
303
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 ...
- 563
2
votes
0
answers
104
views
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
- 647
2
votes
0
answers
524
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 ...
- 15.1k
1
vote
1
answer
102
views
Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
- 851
1
vote
0
answers
65
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 $(...
- 145
1
vote
1
answer
124
views
Why is the doubling dimension of any net of a metric space at most half of that of the metric space?
Definition Doubling dimension ($\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 ...
- 13
0
votes
1
answer
118
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(...
- 1,245
0
votes
0
answers
95
views
Reference Request: Egoroff Theorem for nets
Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?
- 4,969
-1
votes
1
answer
87
views
Pointwise limit of a "net" of measurable functions is measurable? [closed]
Let $(X, \mathcal{A},\mu)$ be a finite measure space with the $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$.
Let $B$ be a separable Banach space. Then, it is well-known from a theorem by Pettis ...
- 2,749
-2
votes
1
answer
135
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 $...
- 59