# Tagged Questions

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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### 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 $X=\mathbb N$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $E$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\beta\varphi$ ...
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### 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 ...
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: ...