LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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A countable entourage base can be replaced in ZF by a decreasing symmetric base whose next triple composite lies in the preceding member

Statement

In ZF, every countably based uniformity has a decreasing symmetric base $(E_{n})$ with $E_{n + 1}^{\circ 3} \subseteq E_{n}$ .

Facts & Assumptions

Given: A countable entourage base $B$ .

[L1]

Symmetric entourages form a base and have square roots (Every uniformity has a base of symmetric entourages).

[L2]

A nonempty subset of $N$ has a least element (The well-ordering principle).

[L3]

Recursion constructs a sequence from a specified starting value and successor map (The recursion theorem).

Proof

technique · constructive

1.1

Use the finite listing or bijection supplied by countability to write the given base as $(C_{n})$ , repeating its last member in the finite case. Put $B_{n} = ⋂_{i \leq n} (C_{i} \cap C_{i}^{- 1}) .$ Then $(B_{n})$ is a canonically defined decreasing symmetric cofinal base.

L1construct

1.2

Define indices recursively. Put $r_{0} = 0$ , and let $r_{n + 1}$ be the least $k > r_{n}$ such that $B_{k}^{\circ 3} \subseteq B_{r_{n}}$ ; then put $E_{n} = B_{r_{n}}$ .

L1L2L3construct

2.1

Each required set of indices is nonempty: choose a symmetric entourage $D$ with $D^{\circ 3} \subseteq B_{r_{n}}$ , then use cofinality and decreasingness to find $k > r_{n}$ with $B_{k} \subseteq D$ . Thus the recursion is defined. The inequalities $r_{n + 1} > r_{n}$ give decreasingness and cofinality, while the defining clause gives triple control.

step 1.1step 1.2L1L2

3.1

Therefore $(E_{n})$ is the asserted normal base in ZF.

step 2.1discharge-construct∎

Depends on

Used by

Every countably based uniformity is generated by one pseudometric, which is a metric exactly when the uniformity is separated Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Wodzicki, Uniform Structure (standard reference, not scraped)
M. Kunzinger, General Topology (standard reference, not scraped)