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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 processing, do not use this tag and instead use (signal-processing) or another appropriate tag.

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Topology filters and convergence

Let $f: X\rightarrow Y$ be a surjective, continous and open mapping, $x\in X$ and $\mathcal F$ a filter on $Y$ such that converges to $f(x)$. Prove that exist a filter $\mathcal C$ on $X$ such that $f(...
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Topology Filters

Let $X$ be a set and $\mathcal F$ a filter on $X$ with a countable filter basis $\mathcal B$. Prove that $\mathcal F$ has a countable filter basis $\mathcal B´$ which is nested, i.e. $\mathcal B´$={$...
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Proof verification: totally bounded iff every filter has a cauchy superfilter

It would be appreciated if someone could check if these proofs are correct (in Schechter's HAF, the 'only if' direction of the theorem below is essentially left as an exercise with a hint, so I wanted ...
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What happens to the definition of a filter if we add a “closed under union property”?

Let $A$ a set. A non-empty subset $\mathcal {F}$ of powerset $\mathcal {P}(A)$ we say filter on the set $A$ if it has the following properties: is upward closed under inclusion, i.e: $X\in {\...
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Singletons in the $\sigma$-algebra generated by clopen sets of a Stone space

Let $A$ be a Boolean algebra and let $Ult(A)$ be its Stone space, that is, the set of all ultrafilters on $A$. It is well known that $C=\{\{u\in Ult(A)\!:a\in u\}\!:a\in A\}$ is an algebra of sets ...
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Does there exist this particular system of subsets of the integers?

Is there a system of subsets of the integers, $\mathcal{E} \subseteq \mathcal{P}(\mathbb{Z})$, such that (B1) $A \in \mathcal{E} \Leftrightarrow A^c \notin \mathcal{E}$ (B2) $A \in \mathcal{E}, A\...
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Duality between Ideal and Filters on a poset

I am studying Ideals and Filters on a poset, and it is clear to me that this are dual notions in the sense that reversing inclusions in the definition we can get the one from the other, or in the ...
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A normal filter including the tail sets is $\kappa$-complete

I need to show that if $\cal F$ is a normal filter on a regular uncountable cardinal $\kappa$, and if $\cal F$ contains all tail sets, i.e. all $$ C_\alpha=\{\beta \ : \ \alpha<\beta<\kappa\...
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Fodor theorem on ultrafilter

Reading some materials on set theory, I know the generalisation of Fodor's theorem: A filter $\cal F$ on a regular uncountable cardinal $\kappa$ is normal if and only if for every regressive ...
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Example of infinite partition of a set $A$, with no elements in ultrafilter on $A$

Let $\mathcal{F}$ be an ultrafilter on a set $A$. It is easy to show that for every finite partition $\{X_n : n<m\}$ of $A$, there exists some $n<m$ such that $X_n\in\mathcal{F}$. Similarly if ...
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Tail set on $\mathcal{F}$ non-principal a $\kappa$-complete ultrafilter on $\kappa$

I'm reading some contents on set-theory for my own interest, and I stumbled upon some questions I cannot solve yet. Let $\mathcal{F}$ be a $\kappa$-complete non-principal ultrafilter on $\kappa$. ...
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$\kappa$-ultrafilter on $\kappa$, creating a partition from $\bigcup_{\alpha < \lambda} X_\alpha \in $ filter

I'm reading some contents on set-theory for my own interest, and I stumbled upon some questions I cannot solve yet. Let $\mathcal{F}$ be a $\kappa$-complete ultrafilter on $\kappa$, and $\bigcup_{\...
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Parameters of Gabor filter

I've designed a program in Matlab using the Gabor filter to recognise handwritten numbers from images. To be more specific, it recognises 4s and 5s. It works pretty well, but I'd like it to be more ...
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Motivation for defining filter convergence

I've just learned about a filter converging in a topological space, but I just can't understand what's the motivation to define such a thing... I get that it is a generalization of a sequence ...
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Counting subsets without a particular element.

Let $W$ be a finite set, and let $\mathcal{P}(W)$ form an algebra. Let $p, q \in \mathcal{P}(W)$. Next, let $S_p = \{s \in \mathcal{P}(W) : p \subseteq s\}$ and $S_q = \{s \in \mathcal{P}(W) : q \...
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Tweak the definition of filter in textbook Introduction to Set Theory 3rd by Hrbacek and Jech

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech defines filter as follows: 1.1 Definition Let $S$ be a nonempty set. A filter on $S$ is a collection $\mathcal F$ of subsets of $S$ ...
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If $\langle a_n \rangle_{n=0}^\infty$ is convergent and $\lim_{n \to \infty}a_n=a$, then for every nonprincipal ultrafilter $U$, $\lim_U a_n=a$

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech introduces a definition related to ultrafilter as follows: 2.7. Definition Let $U$ be an ultrafilter on $\mathbb{N}$, and let $\langle ...
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Closed sets with filters

I am studying topological vector spaces, and, being at the begin of the textbook (Sevres) I frequently encounter rephrasing and generalisations of known results about metric spaces in terms of filters....
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Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....
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Do filters correspond to the collection of their ultrafilter extensions?

Let $X$ be a set, let $\mathscr{F}$ be the collection of filters on $X$, and let $\mathscr U$ be the collection of ultrafilters on $X$. Define $\pi: \mathscr F \to 2^{\mathscr U}$ by $$\pi(F) = \{U \...
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Trace of a filter

Here are the definitions of the extension and trace of a filter (paraphrased from IM James' book "Topological and Uniform Structures"): Let $\mathcal{F}$ be a filter on the set $X$. For each set $X'...
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Restricting a filter in a Boolean algebra to a generating set and have it generate a filter

Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
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Filters and measures

It is well-known that the sets of measure 1 under a given probability measure give rise to a filter. On the other hand, it seems to me (but I'm not certain) that, given a filter, we can define a ...
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Re. Jech Set Theory, Theorem 7.8 ($2^{\aleph_0} = \aleph_1 \implies \exists\: \text{Ramsey ultrafilter}$)

Theorem 7.8 in Jech's Set Theory states that if $2^{\aleph_0} = \aleph_1$, there exists a Ramsey ultrafilter. The proof is constructive: We enumerate all partitions of $\omega$ (denoted $\mathcal{A}_\...
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If $x_n$ is a 0,1 sequence such that $\text{p-lim}(x_n) = 1$, then any 0,1 sequence $y_n \supset x_n$ has $\text{p-lim}(y_n) = \text{p-lim}(x_n) = 1$.

I am working through this post from Terry Tao's blog. Problem: Prove that if a binary (i.e. consisting of $0$ and $1$, or $\top$ and $\bot$) sequence $x_n$ is such that $\text{p-lim}(x_n) = 1$, then ...
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Intuition for the Stone-Čech compactification via ultrafilters

Definitions used: Given some set $X$, denote by $\beta X$ the set of ultrafilters on $X$. We can view $X$ as a subset of $\beta X$ by identifying each point $x \in X$ with the principal ultrafilter ...
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weakly normal filters

Kanamori (Ultrafilters over Uncountable Cardinals) in his Phd Thesis defines a filter $\mathcal F$ as weakly normal whenever every function $f$ such that $\{\xi<\kappa\mid f(\xi)<\xi\}\in\...
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Is there a proof of the proposition that $\equiv$ is an equivalence relation?

Would anyone happen to have a proof that the following use of $\equiv$ is an equivalence relation: Let $\mathcal{F}(\mathbb{N})$ be an ultrafilter on $\mathbb{N}$, constructed by choosing subsets of $...
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Are ultrafilters unique?

I'm trying to get a feel for what an ultrafilter is by looking at some finite examples. Firstly, I took a look at this question: Example of a filter on a set, and it stoked the question: can sets such ...
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Covariance in and input-output filter [Stationary Stochastic Processes]

The weakly stationary processes $X_t$, $\;t=0$, $\,\pm$$1$, $\,\pm$$2$,$\,\ldots$ and $Y_t$,$\;$ $t=0$, $\,\pm$$1$, $\,\pm$$2$,$\ldots$ are input and output of a linear filter according to $...
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Limit point compact uniform space

I'm working on a theorem on compactness for uniform spaces. Here are the definitions I'm using: $X$ is compact if every open cover of $X$ reduces to a finite subcover. $X$ is filter-compact ...
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Determining elements of a Boolean algebra by a set of ultrafilters

Let $A$ be a Boolean algebra and let $Ult(A)$ be its Stone space. Let us say that a set $U\subseteq Ult(A)$ determines an element $a\in A$ if there exists $V\subseteq U$ such that $$\big\{b\in A\!: (\...
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Proof: $U$ is an ultrafilter of a Boolean algebra $B$ if and only if for all $x$ in $U$ exactly one of $x$,$x^*$ belongs to $U$.

I have been stuck with this problem for a while now. I have a proof that letting $U$ be an ultrafilter, exactly one of $x,x^*$ belongs to $U$ for all $x$ in $B$, I did this by showing that both belong ...
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Sequential compactness and filters

I'm trying to work with the many equivalent definitions of compactness for a topological space $X$; in particular, Every (proper) filter on $X$ has a (proper) convergent refinement. I'll ...
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$(f,g) = C(X) \Longleftrightarrow Z(f) \cap Z(g) = \emptyset$

Theorem: a: If $I$is an ideal in $C(X)$ , then the family $Z[I] = \{ Z(f) : f \in I \} $ is a $z$-filter on $X$. b: if $\mathbf{F}$ is a $z$-filter on $X$, then the family $Z^{-1} [ \mathbf{F} ] = ...
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Proving that functions send ultrafilter basis to ultrafilter basis

I'm currently revisiting a proof of Tychonoff's theorem via ultrafilters. The definitions we were working with are as follows, $\mathcal{B}$ is a basis for a filter $\mathcal{F}$ on a set $X$ if $\...
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About isomorphic of a boolean algebra to bolean algebra

Let $\mathcal A$ be a boolean algebra. It`s non-empty subset $\mathcal F$ is called a filter if $∅ \notin \mathcal F$, for all $A ∈ \mathcal A$, $B ∈ \mathcal F$ from $B ⊂ A$ follows $A ∈ \mathcal F$, ...
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find a free ultrafilter on $\Bbb N$

Suppose $(x_n)_n$ is a bounded sequence of complex numbers, there must exist a accumulation point, say $x_0$, thus we can find a free ultrafilter $\mathcal{F}$ on $\Bbb N$ such that $\lim_{\mathcal{F}}...
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intersection of sets corresponding to free ultrafiler

If $\omega \in \beta \Bbb N\setminus \Bbb N$,we define $S_{\omega}=\{(x_n) \in \prod M_n(\Bbb C):lim_{n \to \omega}tr_n(x_n)=0\}$ Is the intersection $\cap_{\omega \in \beta \Bbb N \setminus \Bbb N}...
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difference between convergence along free ultrafilter and common convergence

Suppose $\mathcal{F}$ is any free ultrafilter on $\beta\mathbb{N}\setminus\mathbb{N}$,$(x_n)$ is a sequence of complex numbers. My question is:What is the deference between the limit along $\mathcal{...
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Club filter of $\kappa$ is $\kappa$-complete

I'm trying to show that club filter of $\kappa$ is $\kappa$-complete for uncountable regular cardinal $\kappa$: Let $\kappa$ be uncountable regular cardinal, let $C(\kappa)$ be the club filter ...
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Let $A$ be a Boolean algebra and $F\subseteq A$ be a filter on $A$. Why are the following properties equivalent?

Let $\mathcal{A}$ be a Boolean algebra and $F\subseteq \mathcal{A}$ be a filter on $\mathcal{A}$. Why are the following properties equivalent? $$(1)\,\,\,A\land B\in F\Rightarrow A,B\in F$$ $$(2)\,\,\...
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Countably closed ultrafilters on incomplete Boolean algebras

Suppose that $B$ is a Boolean algebra. Say that an ultrafilter, $U$, on $B$ is countably closed iff whenever $X\subseteq U$ is countable and the meet $\bigwedge X$ exists, $\bigwedge X\in U$. I ...
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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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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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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 ...