Alphabeta Math

Countability Axioms and Cardinal Functions

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✓ 29 results · all verified · 29 also independently AI-judged
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1 · Prerequisites
Combinatorics
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Second countability: an at most countable basis for the topology

Definition

Separability: the existence of an at most countable dense subset

Definition

The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable

Definition

Under choice, weight w(X), density d(X), local character χ(x,X), and character χ(X) as raw cardinal minima and a supremum

Definition

Under choice, Lindelöf degree L(X) and cellularity c(X) as raw cardinal functions

Lemma

Under choice, w(X) is a well-defined cardinal

Lemma

Under choice, d(X) is a well-defined cardinal

Lemma

Under choice, χ(x,X) and χ(X) are well-defined cardinals

Lemma

Under choice, L(X) is a well-defined cardinal

Lemma

Under choice, c(X) is a well-defined cardinal

Corollary

Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold

Lemma

A countable local base can be chosen open and decreasing

Theorem

Every second countable space is first countable

Theorem

Assuming countable choice, every second countable space is separable

Theorem

Assuming countable choice, every second countable space is Lindelöf

Theorem

Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf

Proposition

Second countability is hereditary

Theorem

Assuming countable choice, a countable product of second countable spaces is second countable

Theorem

Assuming countable choice, a countable product of first countable spaces is first countable

Proposition

Every separable space satisfies the countable chain condition

Theorem

Under choice, c(X)≤ d(X)≤ w(X) and χ(X),L(X)≤ w(X)

Proposition

Under choice, for Y⊆ X, w(Y)≤ w(X) and χ(y,Y)≤χ(y,X)

Proposition

Under choice, a continuous surjection does not increase density or Lindelöf degree

Theorem

Under choice, every metrizable space has w(X)=d(X)

Lemma

Under choice, the uncountable Δ-system lemma for finite sets

Theorem

Under choice, every Cantor cube 2^I satisfies ccc

Lemma

Under choice, if |I|>2^ℵ₀, then the Cantor cube 2^I is not separable

False statement

Assuming choice, refuted: every ccc space is separable

False statement

Refuted: every first countable space is second countable

False statement

Refuted: every separable space is second countable

False statement

Refuted: separability is hereditary

False statement

Refuted: Lindelöfness is hereditary

False statement

Assuming countable choice, refuted: Lindelöfness is productive

False statement

Assuming choice and countable choice, refuted: arbitrary products of second countable spaces are second countable

Remark

Implication, preservation, counterexample, and choice ledger for the countability axioms

Sources