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.

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Duals of filters, an explicit formula for meet?

Fix some set $U$. Recall that filters on $U$ are nonempty sets $F$ such that $A\cap B\in F \Leftrightarrow A\in F\land B\in F$. Replacing every element of $F$ with its complement and simultaneously ...
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A filter concentrates on a set

Given a filter $\mathcal F$ on some nonempty set $X$ and some $Y \subseteq X$, people often say that "$\mathcal F$ concentrates on $Y$". Questions: Does this simply mean $$\forall Z \subseteq X ...
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Standard manipulation of Algebraic Riccati equation

I have a problem about Standard manipulation of Riccati equation, the recatti equation is shown as $A_fQ+QA_f^T+\gamma^{-2}QL^TLQ+(D_1-KD_2)(D_1-KD_2)^T+\sum+\delta I=0$ where $\sum$ meets ...
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Generalizing a theorem about filters on a boolean lattice

Let $\mathfrak{A}$ be a bounded distributive lattice with binary meet and join $\sqcap$ and $\sqcup$. I will denote $\partial F = \{ X\in\mathfrak{A} \mid \forall Y\in F: X\sqcap Y\ne \bot \}$ where ...
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An alternate definition of ideals

A filter on a poset if by definition its nonempty subset $F$ such that it does not contain the least element and $A, B \in F \Leftrightarrow \exists Z \in F : (Z \le A \wedge Z\le B)$ for every ...
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Characterization of continuity in terms of filters

Two characterizations of continuity are For all filters $\mathcal{F}$, $\mathcal{F} \to x \implies f(\mathcal{F}) \to f(x)$. $f(\overline{S}) \subseteq \overline{f(S)}$ for all $S$ ...
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Intersection of two filters on a poset

Fix a poset. A filter on the poset is its nonempty subset which is both a down-directed set and an upper set. Conjecture Intersection of two filters is also a filter. I have proved this conjecture ...
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The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that "it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone." And in Jech ...
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Generalization of $f(\overline{S}) \subset \overline{f(S)} \iff f$ continuous

A common characterization of a continuous function $f: X \to Y$ is that for any $S \subset X$, $f(\overline{S}) \subset \overline{f(S)}$. Similarly, closed maps are such that $f(\overline{S}) \supset ...
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Characterization of open and closed maps in terms of filters

Understanding continuity and compactness in terms of filters has been very clarifying for me. Is there a convenient characterization of open and closed maps in terms of filters? For instance, it ...
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Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
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Intuition behind filter on a set

In "Counter-examples in topology" of Steen and Seebach, they define a filter on a set $X$ is a collection F of subsets of $X$ with the following properties: Every subset of $X$ which contains a set ...
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Why is the dual of a filter an ideal?

Jech's set theory, (3rd edition) says that if $F$ is a filter on $S$ Let $I = \left\{ {S - X: X \in F}\right\}$ then $I$ is an ideal of $S$ (dual to $F$). However, let $X,Y \subset S$, $X \in I$ ...
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Characterize R0-space by convergent filters

I want to prove the equivalence of the two following characterizations of R0-spaces. One comes from my textbook (with filters), the other one is taken from wikipedia. First, I will introduce the ...
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Filter of sets containing a subset converges

I'm just learning about filters, and I came across the following exercise in Willard's Topology: Let $X$ be a topological space and $A \subset X$. The cluster points of the filter $\mathcal{F} ...
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Given a finite collection of disjoint subsets of $I$ must every ultrafilter on $I$ contain exactly one?

The title pretty much contains the question, but here's some elaboration: The following is one of the first results one encounters while learning about Ultrafilters. Fact: If $\mathfrak{U}$ is an ...
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Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
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Reducing a double ultrapower to a single ultrapower

I hate having to ask this question, as I know for a fact I have seen the answer before but cannot seem to find it. So I'm breaking down and asking for a reference. Given a structure, let's say a ...
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Ultrafilter closed under negative shift

Does exists an ultrafilter $ \mathcal U $ over $\mathbb N $ such that for every $A \in \mathcal U$ the set $A-1=\{a-1\in \mathbb N : a\in A \land a>1 \} $ is also in $\mathcal U $ ?
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Condition for an Ultrafilter to be Ramsey.

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 ...
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Making sense of the expression $\lim_{x \rightarrow k^+}f(x)$ using filters, and a reference request.

I don't know much filter convergence, so this is addressed to those who do. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote a function. In elementary real analysis, we often write: $$\lim_{x ...
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Every net has an ultranet as subnet: direct proof

I'm currently brushing up my topology using Willard's General Topology. Currently I'm working through the chapters 11 and 12 on nets and filters. Chapter 12 deals extensively with ultrafilters and ...
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pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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Why do we need ultrafilters to make sense of the cartesian product of $\mathcal{L}$-structures

I'm trying understand why we need ultrafilters in model theory. Here is how I see things. Could someone tell me if this is correct ? Further explanations are always welcome. Let $\mathcal{L}$ be a ...
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Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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Literature about Ultrafilters

I am in the early stages of planning my senior project and was wondering if anybody had some recommendations of literature about the applications of ultrafilters in social choice theory, along with ...
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Every nonprincipal ultrafilter on $\omega$ is uncountable.

I imagine this is true as it is easy to prove that any principal ultrafilter on $\mathbb{N}$ is uncountable, and nonprincipal ultrafilters seem in a way bigger. (My proof is $\mathcal{P}(\mathbb{N})= ...
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Proving that every filter is $ℵ_0$–complete

I know that every filter is $ℵ_0$-complete from the definition of being k-complete where k is an infinite cardinal. However I don't know how to prove it specifically.
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Iterated Limits Along an Ultrafilter

Setting: Let $\mathfrak{U}$ be an ultrafilter on an index set $I$. Let $G$ be a compact group with identity $e$, and let $\mathbb{T}$ denote the unit circle in the complex plane. For each $i\in ...
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Connection between ultrafilters and maximal consistent sets

I have been told by reputable sources that ultrafilters over the set of all formulas in a given logic corresponds to a maximal consistent set of formulas in that logic, and I am trying to wrap my head ...
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An elementary proof about filters

I my book draft I have proved a theorem which is equivalent to the following. My proof uses ultrafilters, Galois connections, and the cofinite filter. Let $S$ be a set of filters on some set $U$. ...
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Strengthening of a theorem about filters vs a counter-example

Let $S$ be a non-empty set of filters on a meet-semilattice. If our semilattice is a distributive lattice, then the supremum (on the poset of filters ordered by set-theoretic inclusion) of $S$ is the ...
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Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$ such that $\mathcal{U}\subseteq 2^\mathbb{N}$ is non-principal.

Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$. View $\mathcal{U}$ as a subset of $2^\mathbb{N}$. Question: If $\mathcal{U}$ is non-principal then $\mathcal{U}$ does not have the Baire ...
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Are continuous functions on uniform spaces Cauchy continuous?

Suppose $X,Y$ are uniform spaces. Since uniform structures give topologies $X,Y$ are naturally topological spaces, so we can consider continuous functions $X \to Y$. What I am wondering is if $f ...
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If $U$ is an ultrafilter on $\mathbb{N}$, then $U$ limits exist.

This is rather silly, I expect Asaf will point out what I am missing immediately. Let $U$ be a filter on $\mathbb{N}$. If $\{a_n\}_{n=1}^\infty$ is a sequence of reals, we write $\lim_U a_n = a$ if ...
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When the preimage of the family of all open neighborhoods of a point is cofinal

Let $f \colon X \to Y$ be a continuous map of topological spaces. Denote by $O_y(Y)$ the family of all open neighborhoods of a point $y \in Y$. Define $$ f^{-1}(O_y(Y)) = \bigl\{ f^{-1}(U) \colon U ...
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Non-measurability of ultrafilter on $\omega$

It is well-known that any non-principal ultrafilter on $\omega$ is non-measurable regarded as a subset of $2^{\omega}$. My question is "how well-known" is this fact? Here is the only proof I know: ...
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Prove that $\mathscr F^+ = \mathscr J^+$ and $a \in \mathscr J^+$ iff there exists a filter

Let $\mathscr F$ a filter and let $\mathscr J = \mathscr F^c$ be an ideal dual to $\mathscr J$. Denote $\mathscr F^+ = \{a : (\forall x \in \mathscr F)[a \land x \neq 0]\}$ and $\mathscr J^+ = \{a:a ...
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Distributivity of lattices and prime ideals.

I need help proving that, if $L$ is a lattice with the property that for every nonempty proper filter $F$ and every ideal $I$ such that $F \bigcap I = \emptyset$ then there is a prime filter $G$ such ...
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If a set X has the finite meet property, then there is an ultrafilter such that X is a subset of it.

I need to prove that if $X \subseteq B$ is a set with the finite meet property, then there exists an unltrafilter $U$ of $B$ such that $X \subseteq U$. I know that the finite meet property means that ...
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Stone Representation Theorem

Given two Boolean algebras $A$ and $B$ such that $A$ is a subalgebra of $B$. What is the relation between the Stone space of $A$ and the Stone space of $B$. The question maybe silly but I am getting ...
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About a function ranging filters

Let $U$ be an (infinite) set and $N$ be an (infinite) index set. I denote $\mathfrak{A}$ the set of filters on $U$ (including the improper filter). Let $f$ be an $N$-ary relation that is a set of ...
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Another way to express certain filter

Let $\mathfrak{A}$, $\mathfrak{B}$ be posets, $\mathfrak{A} \subseteq \mathfrak{B}$ (and $\mathfrak{A}$ is the induced order of $\mathfrak{B}$). Let $\mathcal{A}$ be a filter on $\mathfrak{A}$. Help ...
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An example of an ultrafilter

This is Theorem 3.5 (pp. 150) of the book "A course in universal algebra." http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf Theorem 3.15. Let $\bf B$ be a Boolean algebra. (a) ...
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References on filter quantifiers

This post is primarily a reference request. In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is ...
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Ultraproducts and Elementary Embeddings

Let $K= \{A_i: i\in \omega\}$ be a countable collection of $L-$structures. Suppose that for each $A_i, A_j$ in $K$, $\exists A_p \in K$ such that $i,j< p$ and $A_i \prec A_p $ and $A_j \prec ...
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Carlson's translatability - are theses characterisations equivalent?

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, I am trying to show that $$ (\forall I \in \mathcal{I})(\exists I' \in ...
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Net and filter generated by it

Let {s(a)} -such that (a) belongs to order set (A)- is a net from the point of (X) , the net {s(a)} converges to (x) if and only if the filter that generated by it converges to (x)
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Prove that $\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$

Do we have the following identity? $$\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$$ Here $C_i$ is a subset of a set $\Omega$.
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Does ultralimit of sequence change after shift?

Let $(a_n)$ be a bounded sequence of numbers, $\omega$ be an non-principal ultrafilter on $\mathbb N$, then one can assign a limit along ultrafilter $(\omega-)\lim a_n$ to it as is said here. This ...