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Under choice, if $∣ I ∣ > 2^{ℵ_{0}}$ , then the Cantor cube $2^{I}$ is not separable

Statement

Assuming choice, $∣ I ∣ > 2^{ℵ_{0}}$ implies $2^{I}$ is not separable.

Facts & Assumptions

Given: Choice, $∣ I ∣ > 2^{ℵ_{0}}$ , and the product topology on $2^{I}$ .

[L1]

Every nonempty at most countable set is a surjective image of $N$ (A nonempty set is at most countable iff it is a surjective image of $N$ ).

[L2]

The binary sequences have cardinality $2^{ℵ_{0}}$ (Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations, Assuming the Axiom of Choice, $2^{κ} = ∣ P (κ) ∣$ , and Cantor's theorem in cardinal form: $κ < 2^{κ}$ ).

[F1]

A condition on finitely many coordinates defines a basic open cylinder (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).

Proof

technique · contradiction

1.1

Suppose $D \subseteq 2^{I}$ is at most countable and dense. It is nonempty because $2^{I}$ is nonempty, so choose a surjection $s : N \to D$ by [L1]. For each $i \in I$ define its column $c_{i} \in 2^{N}$ by $c_{i} (n) = s (n) (i)$ .

L1assume-contraconstruct

2.1

By [L2] there are only $2^{ℵ_{0}}$ possible columns, whereas $∣ I ∣ > 2^{ℵ_{0}}$ . Thus distinct $i, j \in I$ have $c_{i} = c_{j}$ , which says $d (i) = d (j)$ for every $d \in D$ because $s$ is onto.

step 1.1L2

3.1

The cylinder ${x \in 2^{I} : x (i) = 0, x (j) = 1}$ is nonempty and open by [F1], but step 2.1 makes it disjoint from $D$ , contradicting density. Hence $2^{I}$ is not separable.

step 2.1F1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

35 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

UCR General Topology Notes (standard reference, not scraped)
Cantor cube (Wikipedia) (standard reference, not scraped)

Under choice, if ∣I∣>2ℵ0∣I∣>2ℵ0​, then the Cantor cube 2I2I is not separable