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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
- Analytic and coanalytic sets by closed projection Definition
- Axiom of determinacy for natural-number games Definition
- Cantor sequence space Definition
- Luzin sets, stick, and almost-disjoint guessing at omega one Definition
- Simple continued fractions, convergents, and the integer-coordinate coding of ℕ^ℕ Definition
- Synchronous trees and projection bodies Definition
- The Banach–Mazur category game on sequence spaces and the real line Definition
- The Souslin operation Definition
- A clopen game decided by the first move Example
- Closed subsets of Baire space are tree bodies Lemma
- 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 · two levels
21 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
- 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)