Consider a strictly decreasing infinite (indexed by natural numbers) sequence of sets.
This sequence is a filter base. Can we prove that the filter induced by this base is non-principal?
A filter is principal if it is generated by a single set.
Filter induces by a sequence of sets
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filters
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0A single set is a strictly decreasing sequence of sets. – 2017-01-12
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0@AsafKaragila I pointed explicitly that I consider only infinite sequences – 2017-01-12
1 Answers
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Yes, assuming you mean a sequenced indexed by the natural numbers (or any limit ordinal), such a filter is non-principal. For contradiction, assume the filter consists of supersets of some set $A$. Since $A$ is in the filter, it must itself be a superset of some set $B$ in the original sequence. However, since the original sequence was indexed by a limit ordinal, we know it contains a strict subset of $B$, which is both in the filter but not a superset of $A$.