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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Splitting Frechet filter into two proper filters

Let $\Omega$ be a Frechet filter (=cofinite filter) on an infinite set. Do there exist proper filters $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}\cap\mathcal{B} = \Omega$ and ...
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Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
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Certain constructs on filters and on principal filters

Let $\mathfrak{X}$ be a lattice. I will call a set $S\in\mathscr{P}\mathfrak{X}$ a free star when the least element of $\mathfrak{X}$ is not in $S$ and $X\sqcup Y\in S\Leftrightarrow X\in S\vee Y\in ...
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Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
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Stone-Čech compactification using ultrafilters

Let $X = \omega \cup \{ x \}$ ne the Stone-Čech compactification of $\omega$. (I am viewing $X$ as a subspace of the set of ultrafilters over $\omega$). Let, $\mathcal A$, $\mathcal B$ be two disjoint ...
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Boolean Algebras and Spaces

Show that a countably infinite free Boolean algebra $B$ has a Boolean space homeomorphic to $2^\omega$; where $2$ is the discrete space $\{0,1\}$; hence B is isomorphic to the Boolean algebra of ...
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“Hidden” axiom of choice?

Let $\mu$ be a measure on $S$ such that: $\mu\left(\emptyset\right)=0$ and $\mu(S)=1$ if $X\subseteq Y$, then $\mu(X)\leq\mu(Y)$ $\mu\left(\{a\}\right)=0$ for all $a\in S$ if $X_n$, ...
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Building normal filters around a stationary set

Recently I've been looking at connections between Laver functions on large cardinals and diamonds. While $\diamondsuit$-like principles tend to readily generalize to Laver function-like concepts, I've ...
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Equivalent definitions of P-Points (Ultrafilters)

It is stated in lots of places (including this one) that these two are equivalent definitons for an ultrafilter $u$ to be a p-point. If for every sequence $\left < A_n: n \in \omega \right>$ ...
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Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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Non forking extensions of types as extensions of filters

Given a set of parameters $A$ a type in $S_n(A)$ may be thought of as a maximal filter on the monster model which can be constructed from $A$-definable subsets. Given a type $q\in S_n(B)$ saying that ...
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Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
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An other question about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Conjecture: For every ...
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A problem about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Are the following two ...
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An open ultrafilter converges if and only if it has non-empty intersection?

If $(X,\mathcal T)$ is a topological space then $\mathcal F\subseteq\mathcal T$ is called an open filter on $X$ is (i) $X\in\mathcal F$, $\emptyset\notin\mathcal F$; (ii) $A,B\in\mathcal F$ ...
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Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...
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How generic ultrafilter is used in forcing

So I just learned what ultrafilter is and generic filter is. As all maths are for beginners, it just looks like pure concepts, and I don't see how they are going to be applied in forcing. I am looking ...
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What does it mean for ultrafilter to be $\kappa$-complete?

What does it mean when ultrafilter is said to be $\kappa$-complete? I cannot find suitable Internet resource, so I am asking here.
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EKF to fuse gyroscope and accelerometer readings

I found it interesting to implement EKF for fusing gyroscope and accelerometer data. Trying reach my goal i discovered a lab with some theory explaned, also it has nice app for the phone to stream ...
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Ultrapower and hyperreals

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
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Kalman Filter application to non-linear system.

I want to use the Kalman filter to have a better estimate of the state of a system which I know its equations of motion: ...
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A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
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Weaker notion of first-countability for filters?

While trying to understand the motivation behind the definition of a filter I've stumbled upon the following notion, let's call it "almost first-countability": Let $x$ be a point of a set $X$ and ...
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A totally bounded uniformity and certain filters

Conjecture For every totally bounded uniform space $(U;F)$ and filters $\mathcal{X}$, $\mathcal{Y}$ on $U$ such that $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
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Topology of a convergence space

I am actually having an introduction to filters. Today I was trying to prove that the collection of open sets of a convergence space satisfy the axioms of a topology: O $\subset$ X is open iff $lim ...
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Different uniform spaces having the same set of Cauchy filters

I want to understand how Cauchy space is different than uniform space. For this I need an example: An example of two different uniform spaces having for both of them the same set of Cauchy filters?
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Getting wiener filter coefficients in Matlab

I need to find two coefficients (w1,w2) for a wiener predictor filter of the signal x(n)=0.65x(n-1)-0.7x(n-2)+v(n) where: ...
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Opens in a convergence space

By the book "contemporary mathematics", Beyond Topology (F.mynard , E.Pearl) I am now studying convergence spaces on the book mentioned above. On this book (p.123) I find this definition: A subset O ...
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A boolean algebra is complete iff its Stone Space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
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A relation between certain families of filters and filters on a cartesian product of sets

(By filters, I will mean all filters on a set, including the improper filter.) The product $\mathcal{A}\times\mathcal{B}$ of two filters is the filter defined by the base $\{ A\times B \,|\, ...
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A filterbase generating filter F

Show that a non empty subset $X$ of a filter $F$ in $B$ is a base for $F$ iff $X$ generates $F$ and for all $x,y$ $\in$ $X$ $\exists$ $z $ $\in$ $X$ such that $z$ $\leqq$ x $\wedge$ y.
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Is there a specific term for such collections of filters?

Let $U$ be a set. Concept = "a set $\mathscr{C}$ of proper filters on $U$ such that if $X\in\mathscr{C}$ and $Y$ is a proper filter on $U$ and $Y\supseteq X$, then $Y\in\mathscr{C}$." Is there a ...
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An equivalence relation on filters

Let $F$ be a filter. We say $ X \sim_{f} Y $ iff $X \leftrightarrow Y$ $\in$ $F$. I am able to prove reflexivity and associativity of the relation but not the transitivity. Need help with that. Use ...
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Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also ...
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Minimal Cauchy filters in Cauchy spaces

It is well known that "every Cauchy filter contains a unique minimal Cauchy filter" (Wikipedia) for both metric spaces and uniform spaces (see also this question and answer). Does this theorem ...
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The neighbourhood filter of each point is a minimal Cauchy filter

Wikipedia says (for every fixed uniform space) "The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter." Please help me to prove ...
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Maximal Cauchy filter

Product of two filters $\mathcal{A}$ and $\mathcal{B}$ is defined as the filter $\mathcal{A}\times\mathcal{B}$ generated by the base $$\{A\times B \,|\, A\in\mathcal{A}, A\in\mathcal{B} \}.$$ I call ...
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Does every free filter contain the cofinite filter?

In the answer to this question a free ultrafilter is shown to contain the cofinite filter. But does every free filter contain it too? Obviously the ultrafilter that extend it does.
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Countable intersection on an ultrafilter

Does there exist a nontrivial ultrafilter $a$ such that there are no sequence of sets $K_0,K_1,K_2\dots\in a$ such that $$K_0\cap K_1\cap K_2\dots = \emptyset?$$
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Topology generated by a uniformity

Let $\mathfrak{X}$ be the complete lattice of all filters (including the improper filter) on $U\times U$ (for some set $U$), with the order being the set inclusion of the filters. Consider the ...
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Is it true in general that a filter is given by the intersection of the ultrafilters refining it?

In set theory it holds that any filter is the intersection of all the ultrafilters refining it. By the way the definition of filter can be given in a more general context, that is as a particular ...
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Čech-Stone compactification of $\mathbb N$ and ultrafilters on $\mathbb N$

I have found in the literature that the Čech-Stone compactification $\beta\mathbb N$ of $\mathbb N$ (or more generally, of any discrete topological space) can be identified with ultrafilters on ...
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How to show that a free ultrafilter cannot have an infinite pseudointersection?

The following text is a quote from p.180 of Halbeisen's book Combinatorial Set Theory. This book is also available on website of a course taught by the author. (As mentioned in Asaf's comment, it is ...
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Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
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Non-principal ultrafilters and Ultrafilter Lemma

$\textbf{Definition}$ A family $\mathscr{F}$ of subsets of $I$ has the finite intersection property if for each $S_1, \ldots,S_n\in\mathscr{F}$ it holds that $S_1\cap\ldots\cap S_n\neq\varnothing$. ...
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Any ultrafilter over a finite set is principal

Let $I$ be a finite set. Let $\mathcal{U}$ be an ultrafilter over $\mathcal{P}(I)$. I want to prove that $\mathcal{U}$ is principal. My work: let $\mathcal{U}=\{S_1,\ldots,S_k\}$. Since ...
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Help to understand a proof (about filters)

Niels Diepeveen claims that he has proved the following theorem: Theorem Consider the cofinite filter $F$ on an infinite set. Let $K$ be a collection of ultrafilters such that $\bigcap K=F$. Then for ...
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Applying a Kalman filter to a WiFi power signal

I have created an app that uses the power of a WiFi signal to determine distance to the WiFi access point. Problem with that power reading is that it is not very stable. I have been looking into ...
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The residual field of $\operatorname{Spec}(\prod_{p} \overline{\mathbb{F}_p})$.

Let $I$ be a set and $\mathfrak{U}$ be the set of all ultrafilters on $I$. If $E \subset I$ we define $\mathfrak{U}_E$ to be the set consisting of those elements of $\mathfrak{U}$ which contain $E$. ...
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Find derivation (dB/decade) for given amplitude characteristic of low pass filter [Hz, -]

I am trying to find derivation (differential attenuation) for frequency's 600 and 2000 Hz for given amplitude characteristic of low pass filter, which look like this: I assume, that I should ...