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Questions tagged [filters]

Filters (and ultrafilters) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory. In topology they can be used to study convergence in a more general way than just convergence of sequences. For questions about filters in the sense of signal ...

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free ultrafilter

If $\omega$ is a free ultrafilter on $\mathbb{N}$,$(x_n)$ is a sequence of complex numbers,what is the precise definition of "$lim_{\omega}(x_n)$ does not converge to $x$"?
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How many ultrafilters there are in an infinite space?

I'm stuck with the next exercise from the book Rings of Continuous Functions by Gillman. If $X$ is infinite, there exist $2^{2^{|X|}}$ ultrafilters on $X$ all of whose members are of cardinal $X$. ...
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What does the word “extend” mean in the context of model theory?

Consider the following two problems: (1) Let $L=\{E\}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many ...
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What is a tail filter? [on hold]

I have the definition that suppose F is a filter on N, then a tail filter A is a collection of subsets of N such that there exists a k in N for which (k,infinity) is a subset of A. Does this mean that ...
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Deriving the discrete-time lowpass filter from the Laplace equation

I was trying to demonstrate the discrete-time expression of a lowpass filter: $ y_i = \alpha x_i + (1-\alpha) y_{i-1} $ However my result is completely off the target so I am wondering which of my ...
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Is the algebra map of the ultrafilter monad continuous?

Let $\beta$ be the ultrafilter functor from Sets to Sets, which sends a set $X$ to the set of all ultrafilters on the powerset of $\mathcal{P}(X)$ equipped with its Boolean algebra structure. Then $\...
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28 views

Image of a $z-$ultrafilter is contained in a unique $z-$ultrafilter

I understand that for a continuous map $f:X\longrightarrow Y$, if $\mathcal{F}$ is a $z-$ultrafilter on $X$, then $f(\mathcal{F})=\{A\in Z(Y):f^{-1}(A)\}$ is a prime $z-$filter on $Y$ and may not be a ...
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Ultrafilters preserving infinite joins

A filter $U$ over a boolean algebra $A$ (isomorphic to a powerset algebra) "preserves" a join $a = \bigcup_{i\in I}a_i$, if $a\in U$ implies $a_i\in U$ for some $i\in I$. A join $a$ is infinite if $I$ ...
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Colimits where maps in are determined by maps into a component?

I am curious if there is a class of colimits $\mathsf{colim} D$ where maps $A \to \mathsf{colim} D$ must factor through the colimit cocone $D(X) \to \mathsf{colim}D$ for some $X$? For example, would ...
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When is every principal filter an intersection of ultrafilters?

The question is in the title: what property does a lattice need to have such that for every element of the lattice $x$, there exists a set of ultrafilters in the lattice such that the intersection of ...
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40 views

Show that a prime closed filter is not always a closed ultrafilter

I still don't understand how is that true. Here are the definitions: Let P be a class of closed subsets of a topological space X which is closed under finite intersections and finite unions. A closed ...
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$f : \omega \to \omega$ unbounded on every ultrafilter element, must $f$ be strictly increasing some ultrafilter element?

Consider an ultrafilter $\mathcal{U}$ and $f : \omega \to \omega$ such that $f|_{S}$ is unbounded for every $S \in \mathcal{U}$. Must there exist $S \in \mathcal{U}$ such that $f|_{S}$ is strictly ...
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Nonprincipal ultrafilters over $\mathbb{N}$

So, I'm given $\mathcal{A}\subseteq \mathcal{P}(\mathbb{N})$ that has the property that for any $\mathcal{A}_0\subseteq \mathcal{A}$ finite, $\cap\mathcal{A}_0$ is infinite. I have to show that there ...
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Definition of trace in Bourbaki

Bourbaki, General Topology, p. 61 (1966) What is the definition of trace in the following Proposition? Proposition 8. Let $\mathcal{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the ...
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limit of a sequence along ultrafilter

If $X$ is a compact and Hausdorff topological space,$(x_n)_{n}$ is a sequence in $X$, for any ultrafilter $\mathcal{F}$ on $\mathbb{N}$, I know the fact that $\lim_{\mathcal{F}}x_n$ exists and is ...
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68 views

free ultrafilters on $\mathbb{N}$

Let $\mathbb{N}$ be the set of natural numbers, I know the fact that the number of free ultrafilters on $\mathbb{N}$ is uncountable. I have two questions: 1.How to construct these uncountable ...
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37 views

Filter convergence in discrete space.

Which filters converge in a discrete topological space? Is this correct: Let $(X,\tau)$ discrete topological space and $\cal{F}$ a filter and $\cal{U}_x$ the neighborhood system of $x$. Then $\{x\}$ ...
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What is the difference between a directed set and a filtered category?

This may seem like a stupid question, but these two concepts seem to be identified so often that it's just a detail I've overlooked. Apparently a filtered category is a generalisation of a directed ...
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Topological “filters” and convolutions, are they related or is it just a coincidence of name choice?

Being an engineer, I have sometimes heard of Topological filters and in particular ultrafilters. They seem to have some meaning regarding partial ordering of sets but be of a more pure math origin ...
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Trouble understanding filter and ultrafilter

A filter F on S is a collection of subsets of S in which two conditions hold: If A and B belong to the collection F then A∩B also belongs to the collection. If A belongs to the collection F and A is ...
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Norm on hyperreals ${^*\mathbb R}$ using the ultrafilter construction

Suppose we construct the hyperreals by fixing a free ultrafilter $\mathcal F$ – formalizing the idea of "large subsets of $\mathbb N$" – and defining an equivalence relation between two real sequences ...
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Ultrafilters on finite boolean algebras

I am asked to prove a special case of Stone duality, namely that $B\cong \mathcal{P}(\text{Ult}(B))$ by the map $\phi:B\to \mathcal{P}(\text{Ult}(B))$ given by the homomorphism $$ \phi(x)=\{V\in \text{...
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Example of a free ultrafilter on natural numbers

We know that free ultrafilter exist on natural numbers. I would like to see an example of a free ultrafilter on natural numbers.
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Showing that a net is universal

Let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$ and let $x:\mathcal{U} \rightarrow \mathbb{N}$ be a net such that $x(U) \in U, \forall U \in \mathcal{U}$. Show that $x$ is universal. I tried some ...
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Infinite family $\mathscr{A}\subseteq P(\omega)$ with criteria

First part: Prove that there's an infinite family $\mathscr{A}\subseteq P(\omega)$ such that: $X \in \mathscr{A} \Rightarrow |X|=\aleph_0$ $(X,Y\in \mathscr{A} \wedge X\ne Y)\Rightarrow |X \cap Y|&...
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Why isn't this equation for a notch filter working?

I am trying to put together gain equations for a variety of second order filters. I am using this as a reference: Standard Form for Second Order Filters http://www.kves.uniza.sk/kvesnew/dokumenty/...
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A strictly decreasing nested sequence of non-empty compact subsets of S has a non-empty intersection with empty interior.

S is an Hausdorff topological space. A decreasing nested sequence of non-empty compact subsets of S has a non-empty intersection. In other words, supposing $C_{k}$ is a sequence of non-empty, compact ...
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1answer
49 views

Ultrafilter on a boolean algebra containing $u$ but not $v$

I was reading stone's representation theorem and a part of the proof was omitted and i wanted to check if i got it right: So assume that $B$ is a boolean algebra and $u,v \in B \wedge u\neq v$ and we ...
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62 views

If every finite subset of a theory has an infinite model then the theory has an infinite model

Consider the famous compactness theorem, one of the first theorems in model theory: Let $T$ be a theory, i.e., a set of first-order sentences. If every finite subset of $T$ has a model, then the whole ...
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Is every z-filter $\mathscr{F}$ the intersection of all z-ultrafilters refining it?

Every z-filter $\mathscr{F}$ on a topological space $X$ is the intersection of all z-primefilters refining it. Is it also true that it is the intersection of all z-ultrafilters refining it? Can you ...
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$f(\mathcal F):=\{f(F):F \in \mathcal F\}$ is filter-basis and $G = \{G \subset N: f^{-1}(G) \in \mathcal F\}$

Let $\mathcal F$ be a filter on $M$ and $f:N \rightarrow N$. I want to show that $f(\mathcal F):=\{f(F):F \in \mathcal F\}$ is a filter-basis of a filter $\mathcal G$ on N and that $\mathcal G = \{G \...
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Why does a filter base uniquely define a filter?

Why does a filter base uniquely define a filter? We define the filter base $\mathcal B$ of the filter $\mathcal F$as: $\forall F \in \mathcal F \ \exists \ B \in \mathcal B: B\subset F$ So why can $...
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Can we add a ordinal bigger than every other ordinal and end up with a transitive model?

Given some (set sized) transitive model of ZFC, $M$ we can construct Hyper-$M$ as follows. We construct an ultrafilter $U$ on $Ord_M$ such that $(\exists \alpha.S = \{ \beta : \beta > \alpha \}) \...
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Meaning behind Filter in Set Theory

In a course in logic and set theory, we studied the concept of a Filter. We defined a filter $F \in P(S)$ on $S$ an equivalent of the following definition from Jech's Introduction to Set Theory: (a) $...
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“Freq. response should only be warped between $0$ and $\pi$” $\iff$ “full turn around the unit circle in $z$ equals full turn in $g(z)$”?

Why is it equivalent to say: "freq. response should only be warped between $0$ and $\pi$" $\iff$ "full turn around the unit circle in $z$ equals full turn in $g(z)$"? What does "warped" mean? That it'...
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Minimal filter of a power set.

Let $\mathbb{A} \subset P(A)$ non-empty and $\mathbb{A}$ have finite intersection property, Let $\mathbb{A} \subset F,F' $ such that $F,F'$ are filters and $F = \{ X : \exists T (T \subset \mathbb{A} ...
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What makes Fubini product (tensor product) of filters a natural operation?

In several texts you can encounter tensor product or Fubini product of two filters defined as follows: If $p$ is a filter on a set $X$, $q$ be a filter on a set $Y$, then $$p\otimes q=\{A\subseteq X\...
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Tensor product of ultrafilters corresponds to iterated limit

I will use letters such as $p$, $q$ for filters (or ultrafilters). I want to ask about correspondence between iterated limit and $p\otimes q$-limit, but let me start by mentioning the relevant ...
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Equivalence between conditions for precompact/totally bounded

I'm currently working on a book called "topological vector spaces and distributions. I try to understand the following theorem: Theorem 2 In the proof, there are 2 things I don't understand: 1) In ...
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23 views

Subspace topologies are equal on an equicontinuous subset

In our class, we have already proven that: Let $Y$ is a metric space, $X$ a topological space and $H \subseteq C(X,Y)$ equicontinuous and $\Psi$ a filter base on $H$. Suppose that $pr_x(\Psi) \to ...
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A quotient of a product of prime characteristic fields is isomorphic to a subfield of $\mathbf{C}$

This is an exercise that Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Let $L$ be an infinite set of prime numbers. For every $p \in L$ let $k(p)$ denote a ...
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Alexandroff compactification: continuous function extension

Let $(X, \mathcal{T})$ be a non compact topological space, $\infty \notin X$ and $(X^* := X \cup \{\infty\}, \mathcal{T}^* := \{U \subseteq X^*\mid U \cap X \in \mathcal{T} \land (\infty \in U \...
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Prime filters extending principal filters on the Heyting algebra generated by a preorder

Given a preorder $(X, \preccurlyeq)$ one can define a Heyting algebra $(P_{\preccurlyeq}(X), \cap, \cup, \rightarrow, X, \emptyset)$, where $P_{\preccurlyeq}(X) = \{ A \subseteq X \mid \forall x, y: x ...
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ultrafilter convergence versus non-standard topology

I have recently been reading about the non-standard characterisation of topological spaces, by saying which points of ${^*X}$ are infinitesimally close to which standard points. The theory looks a lot ...
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Is there a Boolean algebra without $\sigma$-complete ultrafilters?

I am looking for an example (in ZFC) of a Boolean algebra $A$ such that no ultrafilter on $A$ is $\sigma$-complete. Since every principal ultrafilter is complete, there has to be no principal ...
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157 views

Products of ultrafilters and products of measures

Suppose $F$ is an ultrafilter on $I$ and $G$ is an ultrafilter on $J$. We let $F\otimes G$ be the collection of subsets of $I\times J$ given by $$F\otimes G=\{X\subseteq I\times J :\{j\in J:\{i\in ...
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Ultrafilter bounded convergent to 0 sequences

Let $\mathscr{F}$ be a free ultrafilter on $\mathbb{N}$. It is known that every bounded sequence $x$ is $\mathscr{F}$-convergent. Hence the space of bounded $\mathscr{F}$-convergent sequences is equal ...
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Prove that closed subspaces of a compact space are compact as well.

Let $(X,\mathcal{T})$ be a compact topological space, let $A \subseteq X$ be a closed subset of $X$, equipped with the subspace topology. Prove that $A$ is compact w.r.t. the subspace topology. ...
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The meet of two filters

Given two filters $\mathcal{F}$ and $\mathcal{G}$ on a set $X$, there's a smallest filter containing $\{F \cup G : F \in \mathcal{F},G \in \mathcal{G}\}.$ (Actually, I can't quite tell if this is a ...
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If $\mathbb{P}\in M$ is atomless and the filter $G$ is $\mathbb{P}$-generic over M, then $G\not\in M$.

[Kunen] If $\mathbb{P}\in M$ is atomless and the filter $G$ is $\mathbb{P}$-generic over M, then $G\not\in M$. So following is the proof by Kunen. 1.) $\mathbb{P}$ \ $G$ = $D$ is dense (This can ...