ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Cylinder sets and continued fractions exhibit the homeomorphism $N^{N} ≅ R ∖ Q$

Example

Under the homeomorphism $N^{N} ≅ R ∖ Q$ , a finite cylinder corresponds to the irrational points in the continued-fraction interval determined by the same finite prefix; extension of prefixes gives nested intervals.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

Baire sequence space $N^{N}$ is homeomorphic to the irrational subspace $R ∖ Q$ . (Baire sequence space is homeomorphic to the irrational real numbers).

[F2]

For simple continued fractions with $a_{0} \in Z$ and $a_{n} \geq 1$ for $n \geq 1$ , the convergents satisfy $p_{n} = a_{n} p_{n - 1} + p_{n - 2}$ and $q_{n} = a_{n} q_{n - 1} + q_{n - 2}$ , with $p_{n} q_{n - 1} - p_{n - 1} q_{n} = (- 1)^{n - 1}$ . A code cylinder $C (a_{0}, \dots, a_{n}) \subseteq N^{N}$ and the real interval $J (a_{0}, \dots, a_{n})$ with endpoints $p_{n} / q_{n}$ and $(p_{n} + p_{n - 1}) / (q_{n} + q_{n - 1})$ are different objects and are not identified; it is the intervals $J$ that are nested as the prefix is extended, with $diam J (a_{0}, \dots, a_{n}) = 1 / (q_{n} (q_{n} + q_{n - 1})) \to 0$ . (Continued-fraction convergents, determinant identities, and nested irrational cylinders).

[F3]

The continued-fraction coding of def-simple-continued-fraction-coding is a bijection from the sequences $(a_{0}, a_{1}, \dots)$ with $a_{0} \in Z$ and $a_{n} \geq 1$ for $n \geq 1$ onto $R ∖ Q$ , and both the coding map and its inverse are continuous for the cylinder and subspace topologies (Infinite simple continued fractions parametrise the irrational real numbers).

Verification

technique · direct

1.1

Work out the first continued-fraction cylinders and show how extending a finite sequence nests the corresponding irrational interval.

givenF2F1

2.1

The parametrisation used here is the specific continued-fraction bijection of [F3], not merely the existence of some homeomorphism, which is all [F1] asserts. By [F3] that bijection and its inverse are continuous for the cylinder and subspace topologies, so cylinder convergence corresponds to ordinary convergence of irrational values; the zero-th coordinate convention is the integer decoding fixed in [F2].

step 1.1F2F1F3

3.1

The preceding construction and implications establish the assertion.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)

Cylinder sets and continued fractions exhibit the homeomorphism NN≅R∖QNN≅R∖Q