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Assuming countable choice, a countable product of first countable spaces is first countable
Statement
Assuming , a countable product of first countable spaces is first countable.
Facts & Assumptions
Given: A point in a countable product of first countable spaces.
Countable choice selects a countable local base in every coordinate (The Axiom of Countable Choice ()).
A product of two at most countable sets is at most countable, and under countable choice a countable union of at most countable sets is at most countable (A product of two at most countable sets is at most countable, Countable unions of at most countable sets, assuming ).
Basic neighbourhoods in the product topology restrict only finitely many coordinates (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, First countable space: a countable neighbourhood base at every point).
Proof
Choose the coordinate local bases by [A1].
Finite-support products of their members form a local base at the given point: refine each of the finitely many restricted coordinates of a basic product neighbourhood by a member of its selected local base.
Finite subsets of the countable index set are countable in total: code the subsets of size at most by -tuples and use finite induction on the product theorem in [L1], followed by the countable-union theorem. For each fixed finite support, the possible coordinate choices are a finite product of countable local bases and hence countable. The union over all finite supports is countable by [L1].
The product is first countable.
Depends on
- First countable space: a countable neighbourhood base at every point
- 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
- A product of two at most countable sets is at most countable
- Countable unions of at most countable sets, assuming $\mathrm{AC}_\omega$
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 63 results over 20 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
- UCR General Topology Notes (standard reference, not scraped)
- First-countable space (Wikipedia) (standard reference, not scraped)