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Baire sequence space and its cylinder topology
Definition
The Baire sequence space is , the set of functions from to itself (The set of all functions ), with the product topology obtained by giving each copy of the discrete topology (The product set of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies). For a finite sequence , its cylinder is . The empty sequence has cylinder , and these cylinders form a basis.
Depends on
- The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- The set $B^{A}$ of all functions $A \to B$
Used by
- Baire sequence space is homeomorphic to the irrational real numbers Corollary
- Simple continued fractions, convergents, and the integer-coordinate coding of ℕ^ℕ Definition
- Infinite simple continued fractions parametrise the irrational real numbers Theorem
- Under Dependent Choice, every nonempty Polish space is a continuous image of Baire sequence space Theorem
- Under the Axiom of Countable Choice, Baire sequence space is Polish, and its standard ultrametric is complete Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 55 results over 24 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)
- Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)