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Normal Moore Spaces, PMEA, and Consistency Strength: Examples

1 · Prerequisites

2 · Summary

Worked examples for the normal Moore space pair. The page computes the stars of the standard metric covers by balls of radius $2^{- n}$ and verifies the two inclusions that make them a development, so that every metric space is a Moore space; and it computes the three-quarter event estimate in the PMEA separation argument, where two good events of measure above three quarters and a two-coordinate difference event of measure one half overlap in a point that separates the two chosen neighbourhoods. The false statement records the relative-consistency refutation of " $Z F C$ proves NMSC": CH yields a normal nonmetrizable Moore space, so assuming $Con (Z F C)$ , $Z F C$ does not prove the normal Moore space conjecture.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-adaptedaudited 2026-09-22Open item page →

Development stars form a countable local base

Example

Let $(X, d)$ be a metric space (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric) with its metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). For $n \in N$ let $G_{n} : = {B (x, 2^{- n}) : x \in X},$ the cover by open balls of radius $2^{- n}$ (Open ball, closed ball and sphere in a metric space). The example computes the stars $St (x, G_{n})$ and verifies that $(G_{n})$ is a development, so that every metric space is a Moore space in the sense of Moore spaces and developments.

Facts & Assumptions

Given: A metric space $(X, d)$ , its balls $B (x, r)$ , and the covers $G_{n} = {B (x, 2^{- n}) : x \in X}$ .

[F1]

Star: $St (x, G_{n}) = ⋃ {B (z, 2^{- n}) : x \in B (z, 2^{- n})}$ , and each $G_{n}$ is an open cover because $x \in B (x, 2^{- n})$ and every metric ball is open (Refinements, locally finite families, point-finite families, and star refinements, Open ball, closed ball and sphere in a metric space, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, claim 1).

[L1]

Metric axioms: $d (x, y) = 0$ if and only if $x = y$ , symmetry, and the triangle inequality $d (x, y) \leq d (x, z) + d (z, y)$ (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[L2]

A set is open in the metric topology exactly when each of its points has a ball inside it, and balls are open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, claim 1).

[L3]

A space carrying its metric topology is metrizable, and every metrizable space is $T_{3}$ , hence regular $T_{1}$ (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, In a metric space every closed set is a zero set and a $G_{δ}$ , and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal, claim 4).

[L4]

For every $ε > 0$ some integer $m \geq 1$ satisfies $1 / m < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L5]

A development is a sequence of open covers whose stars refine every open neighbourhood at each point; a Moore space is regular $T_{1}$ and developable. These stars give a countable local base, also for neighbourhoods that are not open (Moore spaces and developments, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

Verification

technique · direct

1.1givenF1L1

For every $x$ and $n$ one has $B (x, 2^{- n}) \subseteq St (x, G_{n}) \subseteq B (x, 2^{1 - n})$ : the first inclusion holds because $x \in B (x, 2^{- n})$ ; for the second, if $y \in B (z, 2^{- n})$ with $x \in B (z, 2^{- n})$ then $d (x, y) \leq d (x, z) + d (z, y) < 2^{- n} + 2^{- n} = 2^{1 - n}$ by [L1].

2.1step 1.1L2L4L5

Hence $(G_{n})$ is a development. Let $U$ be open and $x \in U$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). By [L2] there is $ε > 0$ with $B (x, ε) \subseteq U$ ; take $m \geq 1$ with $1 / m < ε$ by [L4] and set $n = m + 1$ . Induction gives $2^{m} \geq m$ , so $2^{1 - n} = 2^{- m} \leq 1 / m < ε$ ; then step 1.1 gives $St (x, G_{n}) \subseteq B (x, 2^{1 - n}) \subseteq B (x, ε) \subseteq U$ .

3.1step 1.1step 2.1L2L3L5∎

Consequently every metric space is developable, and [L3] makes it regular $T_{1}$ , so it is a Moore space; the star family ${St (x, G_{n}) : n \in N}$ is the countable local base at $x$ supplied by step 2.1: each star is open as a union of open balls and contains $x$ , and every neighbourhood contains an open neighbourhood to which step 2.1 applies. If $X$ is empty, the covers are empty and the assertions about points are vacuous.

Remarks

The star bound doubles the radius. The lower bound shows the star is not smaller than the ball of radius $2^{- n}$ , and the upper bound shows it is contained in the ball of radius $2^{1 - n}$ ; that two-to-one gap is exactly what makes the development property hold with the factor $2$ .
The same computation works with any null sequence of radii, the powers $2^{- n}$ being chosen only for definiteness.

ExampleConstruction: AI-generatedVerification: AI-generatedaudited 2026-09-22Open item page →

The three-quarter event calculation in the PMEA proof

Example

Let $ν$ be a full extension of a fair-coin product measure on $2^{I}$ (PMEA and PMEA-sigma) and let $E, F, D \subseteq 2^{I}$ be sets with $ν (E) > 3 / 4$ , $ν (F) > 3 / 4$ and $ν (D) = 1 / 2$ , where $D = {z : z (i) \neq z (j)}$ is the difference event of two distinct coordinates $i \neq j$ . The example computes that $ν (E \cap F) > 1 / 2$ and $ν (E \cap F \cap D) > 0$ . This is only the measure-theoretic calculation consumed by the separation lemma; the lemma's additional definitions relate its good events to disjoint neighbourhoods.

Facts & Assumptions

Given: A probability $ν$ on the full power set of $2^{I}$ , sets $E, F$ with $ν (E), ν (F) > 3 / 4$ , and a set $D$ with $ν (D) = 1 / 2$ .

[F1]

Probability and complement: $ν (A^{c}) = 1 - ν (A)$ for every $A$ , and $ν (2^{I}) = 1$ (PMEA and PMEA-sigma).

[L1]

Subadditivity for two or three sets: $ν (A \cup B) \leq ν (A) + ν (B)$ , hence for three sets as well (PMEA and PMEA-sigma).

[L2]

For distinct coordinates $i \neq j$ the difference event has $ν ({z : z (i) \neq z (j)}) = 1 / 2$ (PMEA and PMEA-sigma).

Verification

technique · direct

1.1givenF1L1

$ν (E \cap F) > 1 / 2$ : by [F1] and [L1], $1 = ν (E \cup F) + ν ((E \cup F)^{c}) \leq ν (E) + ν (F) + ν (E^{c} \cap F^{c})$ ; hence $ν (E \cap F) = 1 - ν (E^{c} \cup F^{c}) \geq 1 - (ν (E^{c}) + ν (F^{c})) > 1 - (1 / 4 + 1 / 4) = 1 / 2$ , since $ν (E^{c}) < 1 / 4$ and $ν (F^{c}) < 1 / 4$ by [F1].

2.1step 1.1F1L1L2

$ν (E \cap F \cap D) > 0$ : the complement of the triple intersection is contained in $E^{c} \cup F^{c} \cup D^{c}$ , which has measure at most $ν (E^{c}) + ν (F^{c}) + ν (D^{c}) < 1 / 4 + 1 / 4 + 1 / 2 = 1$ by [L1] and [F1] (using $ν (D^{c}) = 1 / 2$ ). Hence the triple intersection has positive measure and is nonempty.

3.1step 2.1L2∎

Hence there exists $z \in E \cap F \cap D$ ; every such $z$ lies in both $E$ and $F$ and satisfies $z (i) \neq z (j)$ by the definition of $D$ . This example proves no topological conclusion from the abstract sets $E$ and $F$ : in The PMEA three-quarter separation estimate the separately defined good events and separating open sets give that conclusion.

Remarks

Strictness matters. Both good events have measure strictly above $3 / 4$ , so the complement of the triple intersection has measure strictly below $1$ ; with $\geq 3 / 4$ the conclusion $ν (E \cap F \cap D) > 0$ could fail.
Only two coordinates are used, through $ν (D) = 1 / 2$ .

False statementConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-22Open item page →

False: ZFC proves the normal Moore space conjecture

Statement

Relative to $Con (Z F C)$ , it is false that $Z F C$ proves the normal Moore space conjecture: $Z F C$ does not prove that every normal Moore space is metrizable.

Facts & Assumptions

Given: The metatheoretic hypothesis $Con (Z F C)$ and the fixed arithmetization of The standard certified provability predicate.

[F1]

$Con (Z F C)$ implies $Con (Z F C + C H)$ (Positive relative consistency of CH and GCH).

[F2]

Weakening, concatenation, and replacement of proved sentence premises by their proofs preserve derivability (Finite support, weakening, and composition of derivations).

[F3]

$Z F C + C H$ proves that there is a normal nonmetrizable Moore space (CH yields a normal nonmetrizable Moore space, Moore spaces and developments).

[F4]

$N M S C$ is the assertion that every normal Moore space is metrizable. [given]

Refutation

technique · direct

1.1givenF1

Assume $Con (Z F C)$ . Then $Con (Z F C + C H)$ directly by [F1].

2.1step 1.1F2F3F4

If $Z F C$ proved NMSC, weakening would give the same theorem in $Z F C + C H$ . But [F3] gives in that theory a normal nonmetrizable Moore space, contradicting NMSC; by the proof-composition operations of [F2], these two finite derivations concatenate to a $Z F C + C H$ refutation, contrary to [step 1.1].

3.1step 2.1discharge-contradiction∎

Therefore, assuming $Con (Z F C)$ , no such refutation exists and $Z F C$ does not prove NMSC.

Remarks

The consistency hypothesis cannot be dropped. The conclusion is a relative statement; $Z F C$ itself cannot prove $Con (Z F C)$ (The standard certified provability predicate).
The failure is not a theorem of $Z F C$ alone. The witness space is produced under CH; under PMEA every normal Moore space is metrizable (PMEA implies the normal Moore space conjecture), so the statement is independent in the usual relative-consistency sense.

Normal Moore Spaces, PMEA, and Consistency Strength: Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Development stars form a countable local base

Example

Facts & Assumptions

Verification

Remarks

The three-quarter event calculation in the PMEA proof

Example

Facts & Assumptions

Verification

Remarks

False: ZFC proves the normal Moore space conjecture

Statement

Facts & Assumptions

Refutation

Remarks

Sources