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.

learn more… | top users | synonyms

1
vote
1answer
48 views

Non-principal ultrafilters on a set [duplicate]

Let $X$ be a set. If $X$ is finite then all ultrafilters on $X$ are principal, i.e. have the form $\{A \subseteq X : x \in A\}$ for some $x\in X$. But now suppose $X$ is infinite, say $X=\mathbb N$. ...
2
votes
0answers
37 views

Intuition for universal quotient maps [migrated]

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of ...
0
votes
2answers
61 views

Sequential compactness implies compactness: what is wrong with this argument?

Definitions A filter is a poset $(I,\leq)$ such that for any $\alpha,\beta\in I$ there is $\gamma\in I$ such that $\gamma\geq\beta,\gamma\geq\alpha$. A net in a set $X$ is a function from a poset to ...
2
votes
1answer
20 views

Why can't we define intersection with all the elements of a filter?

Let's see the definition of a filter with the "$\subset$" order. Let $X$ be a set. We say a non-empty family $\mathcal{F}$ of subsets of $X$ is a filter if: $\emptyset \notin ...
1
vote
0answers
10 views

How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
1
vote
1answer
69 views

Free filter on Wikipedia

(1) In the second example in Section 3.1 of the Wikipedia article on filter, the last sentence says: A nonprincipal filter on an infinite set is not necessarily free. On the other hand, Martin ...
2
votes
0answers
31 views

What it the intuition behind basis, filters as neighborhoods and neighborhoods?

I know every definition, but as hard as I think I can't get the intuition behind such objects. For example, the definitions I am familiar at most: Take $X$ a topological space with topology $\tau.$ ...
1
vote
1answer
30 views

First-countable uniform space: Is the filter definition of complete equivalent to the sequential definition?

This question is motivated mainly by the topology of first-countable spaces being uniquely determined by their notions of convergence of sequences: the closure of a set $A$ is the same as the set of ...
1
vote
1answer
48 views

Equivalent condition of convergence of Filter

Let $\mathcal F$ be a filter in a topological space $X$. Denote $E_\mathcal F$ the set of elementary filters that are refined by $\mathcal F$. Elementary filters are the filters associated to ...
1
vote
0answers
35 views

Is there a convenient way write sums and products with filters?

If we want to iterate a commutative and associative law of composition over $\{a_i\}_{i \in I}$ in some topological space for an index set $I$, there is a simple way to do it using the directed set of ...
0
votes
1answer
17 views

Convolution bounds

For $t\geq0$, let $g_\beta(t)=e^{-t}\sin(\beta t)$, where $\beta$ is a real number, and for $t<0$, $g_\beta(t)=0$. Find $h*g_\beta(t)$ for all $t\geq0$, where ...
0
votes
1answer
29 views

Convolution of a signal with the butterworth filter.

Let $f(t)$ be a signal that is $0$ when $t<0$ or $t>1$. Show that, for the Butterworth filter, one has $$Ae^{-\alpha t}\int_{0}^{\min\{t,1\}}e^{\alpha\tau}f(\tau)d\tau$$ My attempt: ...
0
votes
0answers
36 views

Is a minimal Cauchy filter bounded?

Let $E$ be a $K$-vector space and $A, B$ two subsets of $E$. We say $A$ absorbs $B$ if there is a $\alpha>0$ such that $B \subseteq \lambda A$ for all $\lambda\in K$ such that $\lambda ...
0
votes
0answers
13 views

A question on Cauchy filters

Let $\mathfrak{F}$, $\mathfrak{G}$ and $\mathfrak{G_1}$ be Cauchy filters in a uniform vector space $(X, \mathfrak{U})$. Let $c\in \mathbb{C}$. (1) If $\lim (\mathfrak{G}- \mathfrak{G}_1) = 0$, can ...
2
votes
1answer
28 views

different definitions of a subnet

The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition ...
1
vote
1answer
51 views

Ultrafilter and upper natural densities

It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies $$d^\ast(A):=\limsup_{n\to +\infty} ...
3
votes
1answer
73 views

Why do ultraproduct structures use a quotient as their universe?

For an $L$-structure $\mathfrak{A}$ with universe $A$, if we have an index set $I$, with an ultrafilter $U$, we create an ultraproduct structure having as its universe $\Pi_I \;A_i/U$. This is the set ...
2
votes
1answer
48 views

Assuming CH, can every tower be extended to a selective ultrafilter (or even a p-point)?

Assume CH. A tower is an almost decreasing family $(A_\alpha)_{\alpha\in\omega_1}$ with no pseudointersection. A selective (also called Ramsey) ultrafilter is one with the property that for every ...
3
votes
1answer
73 views

Characterisation of convergence of bounded sequences via ultra-filters

Let $\{a_n\}_{n\in\mathbb N}$ be a bounded sequence of real or complex numbers and $\mathscr F\subset\mathscr P(\mathbb N)$ be a non-principal ultra-filter. Then $a=\lim_{\mathscr F}a_n$ is ...
0
votes
0answers
3 views

Relation between polynomials and derivatives in Savitsky-Golay fitting

I am seeking a more thorough explanation of some of the properties of Savitsky-Golay (S-G) filters, that are maybe not intuitive to people who have worked with least squares in other contexts. To ...
1
vote
2answers
63 views

Filter without cluster point, then the clopen members have empty intersection

Consider a topological space $(X,\tau)$ and a filter $F$ on $X$ with no cluster point. The set $C$ of all clopen members of $F$ has the finite intersection property. Why has the intersection ...
0
votes
1answer
29 views

PIT implies: In a boolean lattice, every filter can be enlarged to a maximal one

I am working through this proof of Herrlich's Axiom of Choice: $(1)\Rightarrow(2)$: How do you define the quotient lattice $B$ modulo a Filter? And why is the preimage of the maximal filter ...
3
votes
1answer
62 views

How does filtration model information?

Lets say you have a probability space $(\Omega, \mathcal{F},P)$ And a stochastic process on this space $\{X_t, t \in T\}.$ Assume that our process takes vaslues in $\mathbb{R}$. T is a totally ordered ...
1
vote
0answers
59 views

Proof: Every lattice has a maximal filter iff AC

I'm working through a proof of Herrlich's book Axiom of Choice, p.58 (Google books): Equivalent are Every lattice has a maximal filter. Axiom of Choice. In this book, a lattice is ...
5
votes
2answers
110 views

Topological space in which the principal filters are the only filters that converge

Let $(X, \mathcal{T})$ be a topological space in which only the principal filters converge. Show that $\mathcal{T}$ is the discrete topology. It is similar to one of my previous questions (link: ...
1
vote
2answers
55 views

Topological space in which every filter in which every filter converges to every point

Let $(X, \mathcal{T})$ be a topological space in which every filter converges to every element $x \in X$. Show that $\mathcal{T}$ is the trivial topology. I'm kind of stuck on this one for a while ...
0
votes
0answers
51 views

Implementation of the LOWESS-algorithim (local regression data smoothing)

I need to implement the LOWESS-algorithm in a piece of software I am working on. The LOWESS-algorithm is a type of filter, which applies a locally weighted regression on each data point. In this ...
1
vote
0answers
16 views

State space representation for fractional order transfer function

What is the state space representation for the following filters? $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12}$ $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12+1}$ Where $u(t)$ is the input and $y(t)$ is ...
11
votes
3answers
265 views

Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
3
votes
2answers
43 views

How to extends a filter to a convergent ultrafilter?

I am new to the ultrafilters, so I apologise if the question is too elementary. Let S be a collection of sets with the finite intersection property, in a non-compact Hausdorff space. S can be ...
6
votes
1answer
96 views

$M$-amenable ultrafilters on $\kappa$ are $\kappa$-powerset preserving

Let $M$ be a transitive model of $\operatorname{ZFC-}$ and let $$ j \colon M \rightarrow N $$ be elementary with $\operatorname{crit}(j) = \kappa \in \operatorname{wfp}(N)$. Let $U_j$ be defined by ...
3
votes
1answer
65 views

The analogy between two Rudin-Keisler orders

Given a set $X$, an ultrafilter $U$ on $X$, and a function $f\colon X\to Y$, we can push forward $U$ along $f$ to obtain an ultrafilter $f_*U$ on $Y$, defined by $C\in f_*U$ if and only if ...
2
votes
0answers
48 views

Filters and their refinements vs nets and their subnets [duplicate]

True or false? a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter ...
0
votes
1answer
25 views

Proof that no σ-complete non-principal filter on a countable set can exist

Jech (3rd edition, page 73) defines a principal filter as {X : Xo ⊆ X} with Xo a non empty subset of S and (page 77) indicates that there is no non-principal σ-complete (i.e. ω1 complete) filter on a ...
2
votes
1answer
43 views

compactness of the Stone-Čech compactification by ultrafilters

My question is about the proof of compactness of the Lemma 3.1(page 5) in this paper. Let $\beta \mathbb{N}$ be the set of all the ultrafilters on $\mathbb{N.}$ For each $A\subseteq \mathbb{N}$, we ...
0
votes
0answers
34 views

How does the sample space remain constant in filtered

I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like ...
2
votes
2answers
41 views

Ultralimit of an eventually constant generalized sequence

Suppose that $x_{i \in I}$ is a generalized sequence on a compact Hausdorff space $X$, indexed by the directed set $I$, and with the property that $\exists \, j \in I$ such that $\forall i \geqslant ...
1
vote
1answer
31 views

Is a non-principal ultrafilter the same thing as a free ultrafilter?

Can someone please confirm if a non-principal ultrafilter is the same thing as a free ultrafilter. I keep finding conflicting definitions so am not sure.
1
vote
1answer
67 views

Show that every proper filter on a set X can be extended to a proper prime filter?

Are the following enough to complete the proof? The union of a chain of filters is a filter. A maximal filter is an ultra-filter. How I can use Zorn's lemma to find the maximal filter?
-2
votes
1answer
55 views

Showing that the set of all cofinite sets is a filter [closed]

Let $F = \{B \subseteq\mathbb N : \mathbb N\setminus B\text{ is finite}\}$. Show that $F$ is a filter on $\mathbb N$. Let $A$ be a non-empty set. Let $a\in A$. Let $F = \{B \subseteq A : a \in B\}$. ...
1
vote
1answer
31 views

Two ways to describe image of a filter under a function

Let $f$ be a function from a set $A$ to a set $B$. Let $\mathcal{A}$ be a filter on $A$. (Note that I do not require that all filters are proper.) It is easy to verify that $\{Y\in\mathscr{P}B \mid ...
0
votes
0answers
28 views

What is a normal filter on a partially-ordered set?

I'm familiar with the concept of a filter F on a partially-ordered set: a non-empty downward directed (any two elements in F have a common lower bound) upper set (all elements above an element in F ...
6
votes
2answers
125 views

Principal Ultrafilters on natural numbers

Let $E$ be a countable set of subsets of $\mathbb{N} $. Show that the filter generated by $E$ cannot be a non-principal ultrafilter. My idea of solution is: Let $D$ be the filter generated by $E$. ...
1
vote
1answer
40 views

Characterization of the deductive closure of a set of axioms.

Thinking about the seemingly isomorphic nature of theories (I take a theory here as a deductively closed set of sentences) and filters has lead me to ask myself the following question: Can the ...
2
votes
1answer
26 views

Description of filter $F$ generated by a subset $E$ of $\mathcal P(W)$

I am trying to solve exercises from the book Modal Logic by Patrick Blackburn, Maarten de Rijke and Yde Venema. I am having a problem to solve one of the exercises in the section 2.5, the exercise is ...
1
vote
2answers
55 views

Is this use of Zorn's lemma in the proof that every infinite set has a non-principal ultrafilter correct?

I just want to quickly confirm that I am using Zorn's correctly in this short proof. There is a non-principal ultrafilter on any infinite set $X$. Consider the filter $\mathcal{F}$ of of ...
5
votes
0answers
70 views

Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
0
votes
0answers
15 views

Why does multiplying by a wavelet equal to bandpass filtering?

The wavelet transform is given as $ \gamma(s,\tau) = \int f(t) \psi (\frac{t - \tau}{s})dt $ However when it comes to carrying out this transform it turns out that it has a "band pass" effect on ...
3
votes
1answer
93 views

What else does ZFC prove about the “spectrum” of a cardinal number?

An auxiliary definition: Definition 0. Given an infinite set $X$ and a filter $\mathcal{F}$ on $X$, let $\sim_\mathcal{F}$ denote the equivalence relation on $\mathcal{P}(X)$ defined as follows: ...
1
vote
0answers
83 views

Origins of Kalman filter Algorithms in his paper in 1960

Concerning Kalman's original paper published in 1960, "A New Approach to Linear Filtering and Prediction Problems", it seems the majority is to show the orthogonal projection is the optimal estimation ...