# 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.

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### How can I prove the existence of delta-incomplete / countable incomplete ultrafilters?

I am reading a W.A.J Luxembourg paper about nonstandard analysis (https://www.jstor.org/stable/3038221). He presents the following definition I am stuck trying to prove the existence of delta-...
1 vote
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### Different definitions of filters in topology

So I have come across three different definitions for set theoretic filters that are used in topology, all of which only vary in their first axiom. The three different first axioms for a filter over a ...
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### free ultrafilter question

I asked a question here Is this a free ultrafilter, and I'm attempting to write up the solution. $\mathbb{N}=\{1,2,...\}$ and for any $A\subseteq\mathbb{N}$, $A+1=\{a+1:a\in A\}$. Suppose $U$ is a ...
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### Is this a free ultrafilter

Suppose $U$ is a free ultrafilter on $\mathbb{N}$. I'm trying to understand what is the ultrafilter generated by the collection $$G=\{E\subseteq\mathbb{N}: E=A+1, \ for \ some \ A\in U\}$$. I think ...
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### What is the relationship between ultrafilters and propositional theories?

I am learning more about how to use ultrafilters by using them to prove several of the typical results which appear as applications of propositional (or first-order) compactness. Generally speaking, ...
1 vote
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### A question on existence of ultrafilters and cardinality.

Let $S$ be an infinite set with cardinality $k$. Let $U$ be an ultrafilter containing the filter of cofinite sets. Then, for any set $P \subset U$ such that the cardinal of $P$ is $<k$ , is it true ...
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### Dual space of $\mathcal{P}(M)$, when regarded as an $\mathbb{F}_2$-vector space.

This question stems from the well-known observation that the power set $\mathcal{P}(M)$ of any set $M$ can be given an $\mathbb{F}_2$-vector space structure by the symmetric difference operation. Any ...
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### The complement of a filter is an ideal

In the book I'm reading, the author defines a filter like this: Given a non-empty set $X$ a filter over $X$ is a set $f \subseteq P(X)$ such that: (i) $f \not= \emptyset$ (ii) If $S_{1}, S_{2} \in f$ ...
1 vote
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1 vote