# Questions tagged [filters]

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. For questions about filters in the sense of signal processing, do not use this tag and instead use (signal-processing) or another appropriate tag.

589 questions
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### If $x_n$ is a 0,1 sequence such that $\text{p-lim}(x_n) = 1$, then any 0,1 sequence $y_n \supset x_n$ has $\text{p-lim}(y_n) = \text{p-lim}(x_n) = 1$.

I am working through this post from Terry Tao's blog. Problem: Prove that if a binary (i.e. consisting of $0$ and $1$, or $\top$ and $\bot$) sequence $x_n$ is such that $\text{p-lim}(x_n) = 1$, then ...
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### Intuition for the Stone-Čech compactification via ultrafilters

Definitions used: Given some set $X$, denote by $\beta X$ the set of ultrafilters on $X$. We can view $X$ as a subset of $\beta X$ by identifying each point $x \in X$ with the principal ultrafilter ...
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### Are ultrafilters unique?

I'm trying to get a feel for what an ultrafilter is by looking at some finite examples. Firstly, I took a look at this question: Example of a filter on a set, and it stoked the question: can sets such ...
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### Image of a $z-$ultrafilter is contained in a unique $z-$ultrafilter

I understand that for a continuous map $f:X\longrightarrow Y$, if $\mathcal{F}$ is a $z-$ultrafilter on $X$, then $f(\mathcal{F})=\{A\in Z(Y):f^{-1}(A)\}$ is a prime $z-$filter on $Y$ and may not be a ...
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### Ultrafilters preserving infinite joins

A filter $U$ over a boolean algebra $A$ (isomorphic to a powerset algebra) "preserves" a join $a = \bigcup_{i\in I}a_i$, if $a\in U$ implies $a_i\in U$ for some $i\in I$. A join $a$ is infinite if $I$ ...
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### Colimits where maps in are determined by maps into a component?

I am curious if there is a class of colimits $\mathsf{colim} D$ where maps $A \to \mathsf{colim} D$ must factor through the colimit cocone $D(X) \to \mathsf{colim}D$ for some $X$? For example, would ...
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### When is every principal filter an intersection of ultrafilters?

The question is in the title: what property does a lattice need to have such that for every element of the lattice $x$, there exists a set of ultrafilters in the lattice such that the intersection of ...
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### Show that a prime closed filter is not always a closed ultrafilter

I still don't understand how is that true. Here are the definitions: Let P be a class of closed subsets of a topological space X which is closed under finite intersections and finite unions. A closed ...
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### $f : \omega \to \omega$ unbounded on every ultrafilter element, must $f$ be strictly increasing some ultrafilter element?

Consider an ultrafilter $\mathcal{U}$ and $f : \omega \to \omega$ such that $f|_{S}$ is unbounded for every $S \in \mathcal{U}$. Must there exist $S \in \mathcal{U}$ such that $f|_{S}$ is strictly ...
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### Nonprincipal ultrafilters over $\mathbb{N}$

So, I'm given $\mathcal{A}\subseteq \mathcal{P}(\mathbb{N})$ that has the property that for any $\mathcal{A}_0\subseteq \mathcal{A}$ finite, $\cap\mathcal{A}_0$ is infinite. I have to show that there ...
Bourbaki, General Topology, p. 61 (1966) What is the definition of trace in the following Proposition? Proposition 8. Let $\mathcal{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the ...
If $X$ is a compact and Hausdorff topological space,$(x_n)_{n}$ is a sequence in $X$, for any ultrafilter $\mathcal{F}$ on $\mathbb{N}$, I know the fact that $\lim_{\mathcal{F}}x_n$ exists and is ...