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Countable uniformly dense tests on a compact metric space

Statement

Assume AC. For a compact metric K, $C (K; R)$ has a countable uniformly dense subset in the supremum norm.

Facts & Assumptions

[F1]

A compact metric space is complete and totally bounded, and neither implication uses any choice principle: Let $(X, d)$ be a compact metric space (def-metric-compactness, def-metric-space). Then $(X, d)$ is totally bounded (def-totally-bounded) and complete (def-complete-metric-space).

Both implications are theorems of ZF. Completeness is obtained here from the finite intersection characterisation (thm-compact-iff-finite-intersection-property) applied to the closures of the tails of a Cauchy sequence, and not from the extraction of a convergent subsequence, which would route the argument through sequential compactness. What matters for the ledger is that the route taken below selects nothing at all; the first remark below says why the other route was not taken.

[F2]

$∣ d (x, A) - d (y, A) ∣ \leq d (x, y)$ , so the distance to a fixed nonempty set is $1$ -Lipschitz: Let $(X, d)$ be a metric space (def-metric-space), let $A \subseteq X$ be nonempty and let $x, y \in X$ . Then

$∣ d (x, A) - d (y, A) ∣ \leq d (x, y),$

with $d (\cdot, A)$ the distance to a nonempty set (def-metric-bounded-diameter). Thus the real-valued function $u \mapsto d (u, A)$ changes by at most $d (u, v)$ between $u$ and $v$ : it is $1$ -Lipschitz.

[F3]

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous: Let $(X, d_{X})$ be a compact metric space (def-metric-compactness), let $(Y, d_{Y})$ be any metric space (def-metric-space) and let $f : X \to Y$ be continuous (def-metric-continuity). Then $f$ is uniformly continuous (def-metric-uniform-continuity).

No choice principle is used: the cover built below is cut out by a property, and the Lebesgue number lemma it is fed to is itself choice free (thm-lebesgue-number-lemma).

Proof

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1

If K is empty there is one function and the assertion holds. Otherwise F1 supplies finite 1/m-nets. AC chooses these nets with finite listings; their countable union D is dense. Consider all functions $x \mapsto \min_{1 \leq j \leq l} (q_{j} + L d (x, a_{j}))$ , with finite lists $a_{j} \in D$ , rational $q_{j}$ and positive integer L, optionally clipped between rational constants. These form a countable family of continuous functions; F2 with singleton sets gives the needed continuity.

F1F2

1.2

Fix continuous f, $η$ >0 and $M \geq ∥ f ∥_{\infty}$ . By F3 choose $δ$ >0 so $d (x, y) < δ$ implies $∣ f (x) - f (y) ∣ < η$ . Choose integer L with $L δ > 2 M$ . Then $g (x) = \inf_{y \in K} (f (y) + L d (x, y))$ obeys $g (x) \leq f (x)$ by y=x. For d(x,y)< $δ$ the expression is at least f(x)- $η$ ; for d(x,y)>= $δ$ it is greater than -M+2M>=f(x). Thus $f (x) - η \leq g (x) \leq f (x)$ .

2.1

Choose a finite net $a_{1}, \dots, a_{l}$ from D with mesh h< $δ$ and Lh< $η$ , and rational $q_{j}$ with $∣ q_{j} - f (a_{j}) ∣ < η$ . For any y choose $a_{j}$ within h; uniform continuity gives $f (a_{j}) + L d (x, a_{j}) \leq f (y) + L d (x, y) + 2 η$ . Taking the infimum over y and allowing the rational error proves $g (x) - η \leq \min_{j} (q_{j} + L d (x, a_{j})) \leq g (x) + 3 η$ . Together with step 1.2 the error from f is at most 3eta. Rational clipping bounds containing f(K) cannot increase it. Letting $η$ decrease proves density.

step 1.2∎

Depends on

Used by

Probability laws on a compact metric space have weakly convergent subsequences Lemma

Dependency tree · two levels

40 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

van Gaans, Proposition 5.3, pp. 15–16; explicit countable-test replacement for its Alaoglu step (standard reference, not scraped)