Alphabeta Math

Convergence: Nets and Filters: Examples and Counterexamples

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✓ 8 results · all verified · 8 also independently AI-judged
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1 · Prerequisites
Combinatorics
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

A neighbourhood-indexed net in A converges to each point of Ā

Example

Finite partial sums of a real family form a net directed by inclusion

Theorem

Assuming countable choice, a real family is summable as a finite-subset net if and only if it has at most countable support and its nonzero terms are absolutely summable; its sum is independent of the enumeration

Example

Assuming the ultrafilter lemma, a free ultrafilter on ℕ converges to the added point in the one-point convergent-sequence space

Example

The coordinate-reading sequence in a compact binary cube has a convergent subnet but no convergent subsequence

Example

In the cocountable topology on ℝ, a closure point outside [0,1] is reached by a net in [0,1] but by no sequence in [0,1]

Example

The sequential fan is Fréchet–Urysohn and not first countable

Example

Arens space S₂ is sequential but not Fréchet–Urysohn

Sources