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CH yields a normal nonmetrizable Moore space

Statement

$Z F C + C H$ proves that a normal nonmetrizable Moore space exists. By Metrizable spaces are collectionwise normal such a space is the CH instance of the failure of the normal Moore space conjecture.

Facts & Assumptions

Given: The continuum hypothesis $C H$ (The continuum hypothesis, and what this page does not prove): every uncountable set of reals is equinumerous with $R$ ; equivalently here $2^{ℵ_{0}} = ℵ_{1}$ , i.e. $2^{ω} = ω_{1}$ in the von Neumann ordinals (Cardinal (initial ordinal) and cardinality, The natural numbers $N$ (von Neumann)).

[F1]

$ω = ℵ_{0}$ is the least infinite cardinal and $ω_{1} = ω^{+}$ is the least uncountable cardinal; a countable ordinal is one below $ω_{1}$ , and the nonzero limit ordinals below $ω_{1}$ are exactly the countable ordinals of cofinality $ω$ (The natural numbers $N$ (von Neumann), Cardinal (initial ordinal) and cardinality).

[F2]

Clubs and stationarity in $ω_{1}$ : the set of nonzero limit ordinals below $ω_{1}$ contains the club of all limit ordinals and is therefore stationary (Closed unbounded subsets of ordinals, Basic stationary-set calculus).

[F3]

Fleissner's construction (Fleissner's construction of a normal nonmetrizable Moore space from level data): if $κ$ is an infinite cardinal, $(κ_{n})_{n \in ω}$ is an increasing sequence of cardinals with $\sup_{n} κ_{n} = κ$ and $2^{κ_{n}} < κ$ for every $n$ , $2^{κ} = κ^{+}$ , $E \subseteq {δ < κ^{+} : cf (δ) = ω}$ is stationary, and some fixed ladders admit the separation functions $m_{β}$ for every $β < κ^{+}$ , then a normal nonmetrizable Moore space exists (Fleissner's HYP covering interface, Moore spaces and developments).

[F4]

In ZFC, for every at-most-countable set $A$ there is an index set $I_{A}$ which is either a finite initial segment of $ω$ or all of $ω$ , and a bijection $I_{A} \to A$ ; choice permits these bijections to be fixed simultaneously for all $β < ω_{1}$ (The Axiom of Choice, A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain).

Proof

technique · direct; instantiate the construction of [F3]

1.1

Work in $Z F C + C H$ and put $κ : = ω$ , $κ_{n} : = n$ for $n \in ω$ , and $E : = {δ < ω_{1} : δ$ is a nonzero limit ordinal $}$ . Then $E \subseteq {δ < ω_{1} : cf (δ) = ω}$ by [F1], and $E$ is stationary in $ω_{1}$ by [F2].

givenF1F2

2.1

The parameters of step 1.1 satisfy the numerical hypotheses of [F3]: $\sup_{n} κ_{n} = \sup_{n} n = ω = κ$ ; for every $n$ the ordinal $2^{n}$ is a finite ordinal, hence $2^{n} < ω = κ$ ; and $2^{κ} = 2^{ω} = ω_{1} = κ^{+}$ is $C H$ .

step 1.1givenF3

2.2

Fix, for each $δ \in E$ , an increasing sequence $(δ_{i})_{i \in ω}$ of nonlimit ordinals cofinal in $δ$ ; such a sequence exists because $δ$ has cofinality $ω$ , and all the sequences are chosen simultaneously by choice.

step 1.1F4

3.1

For every $β < ω_{1}$ there is a function $m_{β} : E \cap β \to ω$ with $δ_{i} \neq η_{i}$ whenever $δ \neq η$ are in $E \cap β$ and $i \geq \max (m_{β} (δ), m_{β} (η))$ . Indeed $E \cap β \subseteq β$ is at most countable, so by [F4] fix a bijection $k \mapsto δ^{k}$ from an index set $I$ , where $I$ is either a finite initial segment of $ω$ or all of $ω$ , onto $E \cap β$ . For $k < n$ in $I$ the set $F (k, n) : = {i \in ω : (δ^{k})_{i} = (δ^{n})_{i}}$ is finite, because if the two increasing ladders agreed at infinitely many levels then those common values would be cofinal in both $δ^{k}$ and $δ^{n}$ , forcing $δ^{k} = δ^{n}$ . Recursively for $n \in I$ , define $m (n) : = 1 + \max ({0} \cup {m (k) : k < n} \cup ⋃_{k < n} F (k, n))$ . The set maximized over is finite (also when $I$ is finite or empty), so the recursion is well defined. If $k < n$ are in $I$ , then $m (n) > m (k)$ and every $i \in F (k, n)$ satisfies $i < m (n)$ ; hence no coincidence level of the pair reaches $\max (m (k), m (n)) = m (n)$ . Thus $m_{β} (δ^{k}) : = m (k)$ is well defined by injectivity of the enumeration and has the required separation property.

step 2.2F1F4

4.1

Steps 2.1, 2.2 and 3.1 put exactly the hypotheses of [F3] at $κ = ω$ , and ( $κ_{0} = 0$ already) applying it yields a normal nonmetrizable Moore space.

step 2.1step 3.1F3∎

Remarks

The CH instance satisfies the local HYP interface. For every $β < ω_{1}$ one has $cf (β) \leq ω$ , so the interface's clause (3b), which concerns only $cf (β) > ω$ , is vacuous. At $β = ω^{2}$ , the two disjoint sets ${ω n : 1 \leq n < ω}$ and ${ω n + 1 : n < ω}$ are both clubs; thus $E \cap ω^{2}$ contains a club but is nevertheless nonstationary. Step 3.1 independently proves the countable ladder separation that the construction consumes, as in the source's parenthetical countable-case argument.
Where CH is used. Only in $2^{ω} = ω_{1}$ (step 2.1). The stationarity of $E$ and the countable ladder separation are theorems of $Z F C$ .

Depends on

Used by

V=L refutes the normal Moore space conjecture Corollary
False: ZFC proves the normal Moore space conjecture False statement

Dependency tree · two levels

67 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

William G. Fleissner, If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal (standard reference, not scraped)