# Questions tagged [filters]

The filters tag has no usage guidance.

43
questions

**18**

votes

**2**answers

1k views

### Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}...

**8**

votes

**6**answers

978 views

### Spaces of filters

This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better ...

**8**

votes

**0**answers

718 views

### Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...

**7**

votes

**1**answer

690 views

### A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The ...

**6**

votes

**1**answer

199 views

### Reference request: filter tends to filter along map

Recall that a filter on a set $X$ is a nonempty collection $\mathcal{F}$ of subsets of $X$ such that
(i) $U\subseteq V\subseteq X$ and $U\in\mathcal{F}$ implies $V\in\mathcal{F}$, and
(ii) $U,V\in\...

**6**

votes

**1**answer

96 views

### SDE for conditioning on subfiltration

Setup:
Suppose $X_t$ solves the SDE
$$
dX_t = \mu(t,X_t)dt +\sigma(t,X_t)dZ_t,
$$
where $Z_t$ is a Lévy process on $\mathbb{R}^d$, $g(t,s,x):[0,T]\times[0,1]\times \mathbb{R}^d \rightarrow \mathbb{R}^...

**5**

votes

**1**answer

173 views

### Club filter basis in $\omega_1$

My question is about existing of basis of club filter club($\omega_1$) with cardinality $c$. Does it exist?

**5**

votes

**1**answer

398 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)\...

**5**

votes

**1**answer

213 views

### On filters possessing a countable network

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$
A family $\mathcal N$ of subsets of $\omega$ is called a network ...

**4**

votes

**1**answer

177 views

### The property of the dense subfilter of a selective ultrafilter

Let us define the density of subset $A\subset\omega$ :
$$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$
if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...

**4**

votes

**1**answer

170 views

### Convergent filters generated by (not necessarily countable) chains

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}...

**4**

votes

**1**answer

105 views

### The example of the idempotent filter or subsets family with finite intersections property

From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\...

**4**

votes

**0**answers

124 views

### Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...

**3**

votes

**1**answer

112 views

### Dense filter and selective ultrafilter

We say that $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ is the density of subset $A\subset\omega$ if the limit exists. Let us define the filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$.
...

**3**

votes

**0**answers

155 views

### A characterization of Cauchy filters on countable metric spaces?

Given a filter $\mathcal F$ on a countable set $X$, consider the family
$$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$
The following characterization is well-known.
...

**3**

votes

**0**answers

138 views

### Theory of (definable) ideals on a multi-dimensional countable set

I'm wondering if there are any results about definable ideals/filters on $\omega\times\omega$, $\omega\times\omega\times\omega$, etc.
To give a sense of the kind of results I might be looking for: ...

**2**

votes

**2**answers

373 views

### Is certain topology-related set a distributive lattice?

In my research the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...

**2**

votes

**2**answers

261 views

### Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations.
Let $\mathcal{P}(X)$ the power set of $X$.
Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X.
We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...

**2**

votes

**1**answer

168 views

### Interweaving two indexed families of filters

Conjecture
Let $U$ be an (infinite) set. Let $f$ be an $N$-ary (where $N$ is an
arbitrary index set) relation on $U$ (that is a set of functions $N
\rightarrow U$).
Let $\mathcal{L}_0$, $\...

**2**

votes

**1**answer

98 views

### Relationship between wavelet shape and filter points

MATLAB has a library of wavelet functions, showing their "continuous forms" as well as the the decomposition and reconstruction filters.
In decimated wavelet transform the filter size remains the ...

**2**

votes

**1**answer

167 views

### Without Choice: Are there filters of cardinality continuum?

Is it provable, in ZF (without Choice), that every filter can be extended to one of cardinality continuum?
The extended filter is not requested to be an ultrafilter.

**2**

votes

**1**answer

83 views

### Dense subfilter of selective ultrafilter

Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\...

**2**

votes

**0**answers

97 views

### Valuation Rings and Ultrafilters II

See my post here: Valuation Rings and Ultrafilters
Let $K$ be a field, and let $\mathcal{S}$ be the set of pairs $(R, \mathfrak{p})$ of subrings $R$ of $K$ with designated prime ideals $\mathfrak{p}$ ...

**2**

votes

**0**answers

44 views

### Analogous filter to Kalman filter that maximized mode instead (as opposed to minimizing variance)

I may have a potential application where maximizing the mode (as opposed to typically minimizing the variance) would be useful for state estimates. The situation may arise from skewed distributions ...

**2**

votes

**0**answers

293 views

### Research in Algebraic Geometry involving Filters.

In Gillman and Jerrison's book, "Rings of Continuous Functions", they show a nice relationship between the set of z-ideals in C(X) and the set of filters on X. One can go for with this relationship; ...

**1**

vote

**1**answer

395 views

### Do filters complementive to a given filter form a complete lattice?

Really I should first ask this question here on MathOverflow and only then post it as an open problem in Open Problem Garden and propose it as a polymath problem. Indeed I did the reverse and now hope ...

**1**

vote

**1**answer

103 views

### MSE of measurable function is still conditional expectation

Motivation
Then the usual stochastic filtering problem says that:
$$
\operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(Y_t-Z_t)^2],
$$
where $\mathscr{G}_t$ is the $\sigma$-algebra ...

**1**

vote

**1**answer

91 views

### The trace of the filter on a big subset

Let $\scr{F}$ be free filter ($\cap\scr{F}=\emptyset$) on a countable set $X$ and $B\in\scr{F}$. We define the trace of $\scr{F}$ on $B$ as follows $\mathscr{F}_B=\{Y\cap B:~Y\in\scr{F}\}$. $\scr{F}$ ...

**1**

vote

**1**answer

108 views

### Some kind of idempotence of dense filter

In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...

**1**

vote

**1**answer

218 views

### Variance of convolution between filter $A$ and Ornstein-Uhlenbeck process $x_t$

If we consider $x_t$ an Ornstein-Uhlenbeck process (with $W_t$ the Wiener process), does anyone know what would be the variance of the convolution of $x_t$ with a given filter $A$ i.e. $V(x_t \star A)$...

**1**

vote

**1**answer

73 views

### Convergence of a z-filtre to an outer point

Let $X$ be a completely regular topological space and let the set
of all continuous functions from the topological space $X$ into
the topological space $\mathbb{R}$ is denoted by $C(X)$. Let
$Z(X)=\{Z(...

**1**

vote

**1**answer

315 views

### About ordering and equivalence of filters

Let $U$ is a set. I will speak about filters on this set.
If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as
the filter whose base is $\lbrace f[A] | A \in a \rbrace$.
I ...

**1**

vote

**0**answers

30 views

### Posets which extend centered sets to filters

(Post cross-posted from math.se.)
Suppose $(\mathcal O, \leq)$ is an arbitrary poset. Let us say that $\mathcal O$ is compact if every $\mathcal C\subseteq\mathcal O$ which is centered (any finite ...

**1**

vote

**0**answers

124 views

### Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$?

The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:
$$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...

**1**

vote

**0**answers

181 views

### About filters on real numbers

While trying to solve an open problem which I formulated earlier (currently "conjecture 1857" in this file), I met the following problem:
Let $\Delta_{\geq}+\alpha$ be the filter on $\mathbb{R}$ ...

**1**

vote

**1**answer

162 views

### Digital Filters [closed]

Can somebody help with the constructing filter by amplitude and phase spectrum? Is it possible?
I try to google it, but unsuccessufully.
I have some thoughts about solving it by system of linear ...

**0**

votes

**2**answers

1k views

### Algorithm for calculating the sum-of-squares distance of a rolling window from a given line function

Given a line function $y = ax + b$, it is easy to calculate the sum-of-squares distance between the line and a window of samples $(1, y_1), (2, y_2), ..., (n, y_n)$ (where $y_1$ is the oldest sample ...

**0**

votes

**0**answers

16 views

### Magnitude spectrum of a cascade of filter

Given is a input vector $x=[x_1 x_2 x_3] \in \mathbb{R}^{3N}$ with 3 consecutive sub-blocks $x_1,x_2,x_3 \in \mathbb{R}^{N}$, which goes through a cascade of filtering operations defined as
Step 1 (1-...

**0**

votes

**0**answers

188 views

### $\kappa$-translatability

I asked the following on MSE a few weeks ago but I did not get any answer :
https://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent
Reference ...

**-1**

votes

**1**answer

235 views

### Lowering from filters to ultrafilters for an infinitary relation

Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$).
I denote $\mathcal{L}\in \upuparrows f \...

**-2**

votes

**1**answer

292 views

### Expressing a value related to an infinitary relation through ultrafilters

Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and ...

**-9**

votes

**1**answer

1k views

### Filters and intersection of two binary relations

Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} =...

**-12**

votes

**1**answer

2k views

### Direct product of filters

Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.
I will denote the principal filter ...