Normality in generalized Rudin spaces

Introduction

In her 1971 construction [14] of a ZFC Dowker space¹, Mary Ellen Rudin considered two related subspaces of the box product $\prod_{0<n<\omega}(\omega_n+1)$:

and its subspace

She proved that the space $X$ is normal (even collectionwise normal) but not countably paracompact. By Dowker’s classical result [2] it follows that $X$ is a normal space for which $X \times[0,1]$ is nonnormal, that is, a Dowker space.

These two spaces are closely connected. For example, it is easy to see that $X$ is a dense subspace of $X'$ and Rudin herself mentions that $X'$ is the Hewitt realcompactification of $X$. More important for us, $X'$ possesses strong separation properties that can be exploited to prove things about $X$. This was explicitly noted by Hart [8], who points out that if we take a discrete family $\mathcal{H}$ of subsets of $X$, then the family $\overline{\mathcal{H}}$ of their closures in $X'$ will be discrete in $X'$. The space $X'$ is paracompact (even ultraparacompact) and hence collectionwise normal, and an expansion of $\overline{\mathcal{H}}$ to disjoint open sets in $X'$ gives us such an expansion of $\mathcal{H}$ in the space $X$. Thus, $X$ inherits its collectionwise normality from $X'$.

Rudin’s space remained the only known example in ZFC of a Dowker space for many years, but more examples were discovered in the 1990s: in 1996 Balogh [1] constructed a Dowker space of size $2^{\aleph_0}$, while Kojman and Shelah [13] used the existence of a scale at $\aleph_\omega$ to show that Rudin’s space contains a closed Dowker subspace of (absolute) cardinality $(\aleph_\omega)^{\aleph_0}$.²

Recently, Kojman [12] generalized Rudin’s construction to larger cardinals. This translation replaces $\aleph_0$ by an arbitrary infinite cardinal $\mu$ and replaces the sequence $\langle\aleph_n : n < \omega\rangle$ by a strictly increasing sequence of regular cardinals $\bar{\kappa} = \langle\kappa_n : n < \omega\rangle$ greater than $\mu$. The resulting space $X(\mu,\bar{\kappa})$ turns out to be a Dowker space provided the cardinal $\delta= \sup\bar{\kappa}$ satisfies a certain pcf-theoretic assumption. This assumption holds automatically when $\delta$ is a singular strong limit cardinal of cofinality $\omega$, and so this construction answered a question of Rinot by demonstrating that the cardinalities of Dowker spaces form a proper class in ZFC.

The study of these generalized Rudin spaces is continued in [4], where it is shown that the resulting spaces depend only on the choice of $\mu$ and $\delta= \sup\bar{\kappa}$, as different cofinal sequences of regular cardinals above $\mu$ give homeomorphic spaces. Thus, if $\mu< \delta$ are infinite cardinals and $\delta$ has countable cofinality, we can unambiguously define the generalized Rudin space $X(\mu,\delta)$ to be the homeomorphism type of

with the topology inherited from the box product, where $\bar{\kappa}$ is any $\omega$-sequence (not necessarily increasing or one-to-one) of regular cardinals from the interval $(\mu,\delta)$ with supremum $\delta$. With this notation, Rudin’s original Dowker space $X$ is the special case $X(\aleph_0,\aleph_\omega)$, and the main results of [12], together with the pcf reformulation established below, tell us

• $X(\mu,\delta)$ is never countably paracompact, and

• $X(\mu,\delta)$ is normal (hence a Dowker space) if

Here “pp” refers to Shelah’s pseudopower function. For a singular cardinal $\nu$ of countable cofinality, $\operatorname{pp}(\nu)$ is the supremum of the cofinalities $\operatorname{cf}(\prod A/U)$, where $A \subseteq\nu\cap\operatorname{Reg}$ is countable and cofinal in $\nu$ and $U$ is an ultrafilter on $A$ disjoint from the ideal of bounded subsets of $A$. If $\delta$ is a strong limit, then $\operatorname{pp}(\nu) \leq\nu^{\aleph_0} \leq2^\nu< \delta$ for every $\nu\in(\mu,\delta)$ of countable cofinality, so condition (2) holds.

Condition (2) holds vacuously for every space $X(\mu,\mu^{+\omega})$, since each cardinal strictly between $\mu$ and $\mu^{+\omega}$ is a successor. In particular, Rudin’s space $X(\aleph_0,\aleph_\omega)$ satisfies (2) even when $2^{\aleph_0} > \aleph_\omega$. Moreover, assumptions such as GCH or the weaker Shelah Strong Hypothesis³ guarantee that every generalized Rudin space $X(\mu,\delta)$ will be normal and hence Dowker.

The smallest possible counterexample to normality is $X(\aleph_0,\aleph_{\omega+\omega})$, for which condition (2) reduces to

By contrast, $X(\aleph_\omega,\aleph_{\omega+\omega})$ satisfies condition (2) vacuously.

It is therefore natural to ask whether normality must hold in ZFC even when condition (2) fails, and we are led to the following questions left open by [12] and [4]:

Main Question Is every Rudin-type space $X(\mu,\delta)$ normal? If not, can the space $X(\aleph_0,\aleph_{\omega+\omega})$ fail to be normal?

In this paper, we answer both of these questions by showing that condition (2) is also necessary for $X(\mu,\delta)$ to be normal. More precisely, we prove the following characterization of normality in terms of pcf theory:

Given the above theorem, we can answer both parts of our main question by quoting known consistency results to show that the smallest previously unresolved case can indeed fail to be normal.

Any model of ZFC in which $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}>\aleph_{\omega+\omega}$ satisfies $\operatorname{pp}(\aleph_\omega)>\aleph_{\omega+\omega}$, so $X(\aleph_0,\aleph_{\omega+\omega})$ is not normal by Theorem 1.1. In 1983 Shelah [15, Theorem 2.2(1)] constructed such models from a supercompact cardinal. Gitik and Magidor [7, 6] later gave a different construction using extender-based forcing and also arranging for GCH to hold below $\aleph_\omega$, something that Shelah’s construction did not achieve.

The strong limit assumption is not necessary for the failure of normality, however, because adding Cohen reals to such a model will preserve the inequality $\operatorname{pp}(\aleph_\omega)>\aleph_{\omega+\omega}$. This shows that the space $X(\aleph_0,\aleph_{\omega+\omega})$ can fail to be normal in models in which the continuum is arbitrarily large.

Finally, the pcf criterion yields topological characterizations of normality in terms of cardinal invariants. In § 4, we compute the global pseudocharacter $\psi(X)$, character $\chi(X)$, and tightness $t(X)$ of these spaces. Together with Theorem 1.1, these calculations give the following corollary.

In fact, $\psi(X)=\delta$ and $\chi(X)=t(X)$ always hold for these spaces. Thus failure of normality is witnessed by the strict inequality $\psi(X)<\chi(X)=t(X)$.

Notation and preliminaries

Our notation is standard, so in this section we will only sketch the concepts we need and give references to the literature.

Products and Rudin-type spaces

We order ordinal-valued functions pointwise, with $f<g$ meaning $f(i)<g(i)$ at every coordinate, and define

For a set $A$ of regular cardinals, write $\prod A=\prod_{\kappa\in A}\kappa$ and let $\operatorname{cf}(\prod A,<)$ denote the least size of a pointwise cofinal subset.

Let $\mu$ be an infinite cardinal and $\vec{\kappa}=\langle\kappa_n:n<\omega\rangle$ a sequence of regular cardinals above $\mu$ with no largest term. The terms may repeat and may occur in any order. Put $\delta=\sup_n\kappa_n$. Define $X(\mu,\vec{\kappa})$ by (1) and put

Both spaces inherit the box topology from the order topologies. In either space, intersecting $(h,x]$ with the space, for $h<x$, gives a neighborhood base at $x$.

For a countably infinite set $C$ of regular cardinals above $\mu$ with no maximum, $X(\mu,C)$ and $X'(\mu,C)$ denote the corresponding $C$-indexed spaces. By [4], the homeomorphism type of the pair $(X'(\mu,\vec{\kappa}),X(\mu,\vec{\kappa}))$ depends only on $\mu$ and $\delta$, giving the notation $X(\mu,\delta)$.

Topological conventions

For $y\in Y$, let $\chi(y,Y)$ denote its character and $\psi(y,Y)$ its pseudocharacter. Thus $\chi(y,Y)$ is the least size of a neighborhood base at $y$, and $\psi(y,Y)$ is the least size of a family of open neighborhoods whose intersection is $\{y\}$. The tightness $t(y,Y)$ is the least infinite cardinal $\kappa$ such that, whenever $y\in\overline{D}$ for $D\subseteq Y$, there is $E\subseteq D$ with $|E|\leq\kappa$ and $y\in\overline{E}$. We take these invariants to be infinite and put

Pcf preliminaries

Let $J$ be a proper ideal on a set $A$ of regular cardinals. For $f,g\in\prod A$, define

For the ideal $\operatorname{Fin}$ of finite subsets, we write $<^*$ and $=^*$. The ideal of subsets of $A$ bounded in $\sup A$ is denoted by $J^{\mathrm{bd}}[A]$. If $A$ has order type $\omega$, then $J^{\mathrm{bd}}[A]=\operatorname{Fin}$.

If $\lambda$ is regular and $\prod A$ has a $<_{J}$-increasing, $<_{J}$-cofinal sequence $\langle f_\alpha:\alpha<\lambda\rangle$, we write

and call the sequence a $\lambda$-scale. The true cofinality need not exist. For an ultrafilter $U$, the notation $\prod A/U$ uses the ideal dual to $U$.

A nonempty set $A\subseteq\operatorname{Reg}$ is progressive if $|A|<\min A$. For such a set, put

The pcf theorem [5 Proposition 2.19] gives a largest member of $\operatorname{pcf}(A)$ and the identity

Also, $A \subseteq B$ implies $\operatorname{pcf}(A) \subseteq\operatorname{pcf}(B)$. Our applications concern nonempty countable sets of uncountable regular cardinals, which are progressive. We refer to [16, 10] for these facts.

We use pseudopowers only at singular cardinals of countable cofinality. For such a cardinal $\nu$, let $\operatorname{PP}(\nu)$ consist of the regular cardinals of the form $\operatorname{tcf}(\prod A/J)$, where $A \subseteq\nu\cap\operatorname{Reg}$ is countable and cofinal in $\nu$, and $J$ is a proper ideal on $A$ containing $J^{\mathrm{bd}}[A]$. Then

We use the following facts; see [16 Chapter II].

1. There is a cofinal set $A \subseteq\nu\cap\operatorname{Reg}$ of order type $\omega$ such that $\operatorname{tcf}(\prod A/\operatorname{Fin})=\nu^{+}$. In particular, $\nu^{+} \in\operatorname{PP}(\nu)$.

2. If $\lambda\in\operatorname{PP}(\nu)$ and $\nu< \theta< \lambda$ is regular, then $\theta\in\operatorname{PP}(\nu)$ (No Holes).

3. If $\theta< \eta\leq\operatorname{pp}(\theta)$ and $\operatorname{cf}(\theta)=\operatorname{cf}(\eta)=\aleph_{0}$, then inverse monotonicity [3 Proposition 1.4.3] gives $\operatorname{PP}(\eta) \subseteq\operatorname{PP}(\theta)$ and hence $\operatorname{pp}(\eta) \leq\operatorname{pp}(\theta)$.

We finish with the normality criterion from [12].

By (3), the hypothesis can also be written $\max\operatorname{pcf}(A)\leq\kappa_n$.

The pcf characterization of normality

This section will establish Theorem 1.1. We start with some easy lemmas that show us that the pcf-theoretic content of the theorem can be formulated in a few equivalent ways.

Moving on to the statement of Theorem 1.1, we make use now of the countable cofinality of $\delta$. The first two clauses of the following lemma express the negations of items (2) and (3) of Theorem 1.1, respectively, since $\max\operatorname{pcf}(A_0)$ is regular and $\delta$ is singular. We will use the two scales at the end of the lemma to prove normality.

Proof. By Shelah’s scale theorem [16 Chapter II], fix a cofinal set $B_{0} \subseteq\delta\cap\operatorname{Reg}$ of order type $\omega$ such that

(1) $\Rightarrow$ (2). Choose an ultrafilter $U$ on $A_{0}$ witnessing $\lambda= \max\operatorname{pcf}(A_{0})$, and let $\nu$ be the least ordinal with $A_{0} \cap\nu\in U$. As in the proof of Lemma 3.1, $\mu< \nu\le\sup A_{0}$, $\operatorname{cf}(\nu) = \aleph_{0}$, and the restriction of $U$ to $A_{0} \cap\nu$ witnesses $\lambda\in\operatorname{PP}(\nu)$. Thus $\operatorname{pp}(\nu) \ge\lambda> \delta$, and $A_{0} \cap\nu$ witnesses (1) with supremum $\nu$.

(2) $\Rightarrow$ (3). Since $\nu< \delta\le\operatorname{pp}(\nu)$ and both $\nu$ and $\delta$ have countable cofinality, inverse monotonicity [3 Proposition 1.4.3] gives

The scale on $B_{0}$ witnesses $\delta^{+} \in\operatorname{PP}(\delta)$, so the same $\nu$ witnesses (3).

(3) $\Rightarrow$ (4). Choose a countable cofinal set $A \subseteq\nu\cap\operatorname{Reg}$ and a proper ideal $J \supseteq J^{\mathrm{bd}}[A]$ witnessing $\operatorname{tcf}(\prod A/J) = \delta^{+}$. Discarding the bounded initial segment of $A$ at or below $\mu^{+}$ and restricting $J$ accordingly, we may assume $\mu^{+} < \min A$. Let $B = \{b \in B_{0} : b > \nu\}$. This removes only finitely many terms from $B_{0}$, so the restricted scale still has length $\delta^{+}$, and $\sup A = \nu< \min B$.

For each $\kappa\in A \cup B$ and $\xi< \kappa$, put

where the product is ordinal multiplication. Since $\kappa> \mu^{+}$, $e_{\kappa}$ is a strictly increasing map from $\kappa$ to $\kappa$ with $e_{\kappa}(\xi) > \xi$ and $\operatorname{cf}(e_{\kappa}(\xi)) = \mu^{+}$. Replacing each scale function $f$ by the function $\kappa\mapsto e_{\kappa}(f(\kappa))$ therefore preserves strict increase modulo the respective ideal and preserves cofinality.

(4) $\Rightarrow$ (1). Any ultrafilter on $A$ disjoint from $J$ preserves the scale on $A$. Thus $\delta^{+} \in\operatorname{pcf}(A)$, and $A_{0} = A$ witnesses (1). □

The preceding lemma tells us that the two pcf conditions in Theorem 1.1 are equivalent, so what remains is to prove that these give necessary and sufficient conditions for the associated generalized Rudin space to be normal.

Sufficiency:

Assume (3) of Theorem 1.1, so $\operatorname{pp}(\nu) < \delta$ for every cardinal $\nu\in(\mu,\delta)$ of countable cofinality. By Lemma 3.1, a regular $\kappa\in(\mu,\delta)$ satisfies Kojman’s bound precisely when $\operatorname{pp}(\eta) \le\kappa$ for every cardinal $\eta\in(\mu,\kappa)$ of countable cofinality. We show that such $\kappa$ are cofinal in $\delta$.

Fix a regular cardinal $\gamma\in(\mu,\delta)$. If $\operatorname{pp}(\eta) \le\gamma$ for every cardinal $\eta\in(\mu,\gamma)$ of countable cofinality, take $\kappa= \gamma$. Otherwise, let $\theta$ be the least cardinal of countable cofinality in $(\mu,\gamma)$ with $\operatorname{pp}(\theta) > \gamma$, and put

By (3), $\tau< \delta$, and hence $\kappa< \delta$ since $\delta$ is a limit cardinal. For $\eta\in(\mu,\theta)$ of countable cofinality, minimality gives $\operatorname{pp}(\eta) \le\gamma$. For $\theta< \eta\le\tau$ of countable cofinality, inverse monotonicity gives $\operatorname{pp}(\eta) \le\tau$. Thus $\kappa$ works in this case as well.

As $\gamma$ was arbitrary, we can choose a strictly increasing sequence $\overline{\kappa} = \langle\kappa_n : n < \omega\rangle$ of such regular cardinals cofinal in $\delta$. By Theorem 2.1, $X(\mu,\overline{\kappa})$ is normal, and its homeomorphism type is $X(\mu,\delta)$, as required.

Necessity:

We prove (1) $\Rightarrow$ (3) of Theorem 1.1 by contraposition. The model for our proof is the classic example (due independently to Dieudonné and Morse) of the product of ordinal spaces $(\omega_1+1) \times\omega_1$. In their setting, the vertical edge $\{\omega_1\} \times\omega_1$ and the diagonal are disjoint closed sets. An application of Fodor’s lemma then shows that every open neighborhood of the diagonal contains a final square $(\beta,\omega_1)^2$, and that every open neighborhood of the edge meets this square. In our setting, we will form a diagonal by pairing points with the same index on the two scales from Lemma 3.2, and then interleave two sequences with a common pointwise supremum, as in Kelley’s treatment of the ordinal example [9 Chapter 4, Problem E]. For related pressing-down arguments in products, see [13, 11].

Proof. Put $\lambda= \delta^{+}$. By Lemma 3.2(4), choose $A$, $B$, $J$ and scales

whose coordinate values all have cofinality $\mu^{+}$. The first scale is increasing and cofinal in $\prod A$ modulo $J$, and the second is strictly increasing and cofinal in $\prod B$ modulo the finite ideal.

Put $C = A \cup B$ and use $X = X(\mu,C)$ as our presentation of $X(\mu,\delta)$. Let $\Delta(c) = c$ be the diagonal function for $c \in C$. Then $\Delta\in X'(\mu,C) \setminus X$, since the cofinalities of its coordinates are unbounded in $\delta$.

Our analogue of the vertical edge is the closed set

Indeed, $F$ is the intersection, over $a \in A$, of the preimages of the closed singletons $\{a\} \subseteq a+1$ under the continuous coordinate projections $x \mapsto x(a)$. Replacing the $A$-coordinates of any $x \in X$ by their top values leaves their cofinalities bounded by $\sup A < \delta$, so $F$ is cofinal in $(X,\le)$. On the other hand, every $x \in X$ satisfies $x(b) < b$ for all but finitely many $b \in B$, since $B$ has order type $\omega$ and is cofinal in $\delta$.

For our analogue of the diagonal, we put $r_\alpha= p_\alpha\cup q_\alpha$, and define

Thus, membership in the set $D$ is unchanged if we make a small modification to an element, provided we keep control of the cofinalities: on the coordinates in $A$ we are allowed to modify $r_\alpha$ on a $J$-small set, while we are allowed to make finite changes on the coordinates from $B$. The common cofinality bound gives $D\subseteq X$, and we define

The closed subsets $F$ and $G$ of $X$ are intended to refute the normality of the space. The next claim tells us that these are indeed disjoint closed subsets of $X$:

Our next step is to show that every open neighborhood of the closed diagonal $G$ contains an end segment of $X\cap\prod C$.

Finally, we show that every open neighborhood of our set $F$ meets every end segment of $X\cap\prod C$.

Proof. Let $V$ be an open neighborhood of $F$ and fix $h \in\prod C$. Define $y \in F$ by $y(a)=a$ for $a \in A$ and $y(b)=h(b)+\mu^{+}$ for $b \in B$. Choose $g<y$ with $(g,y]\cap X\subseteq V$ and set

Every coordinate of $z$ has cofinality $\mu^{+}$, so $z\in X\cap\prod C$. Since $h<z$ and $g<z\leq y$, this $z$ belongs to $V$ and to the required tail. □

The conjunction of Claims 3.3.2 and 3.3.3 tells us that every open neighborhood of $F$ must meet every open neighborhood of $G$, and therefore the disjoint closed sets $F$ and $G$ witness that $X$ is not normal. □

This proves (1) $\Rightarrow$ (3) and completes the proof of Theorem 1.1. In the construction above, both $F$ and $G$ have $\Delta$ in their closure in $X'(\mu,C')$, as witnessed by the points $y$ in the last claim and by the cofinality of $D$, respectively. The open neighborhood $X\setminus G$ of $F$ misses $D$, so it contains no end segment of $X\cap\prod C$. For comparison with the end-segment property in Rudin’s original space, see [11 Lemma 2.25].

Topological characterization of normality

In the preceding section we showed that $X(\mu,\delta)$ is normal precisely when $\max\operatorname{pcf}(A)<\delta$ for every nonempty bounded countable set $A$ of regular cardinals in $(\mu,\delta)$. We now show how this condition has a direct translation in terms of cardinal functions defined from the topology of $X(\mu,\delta)$. For each such $A$, there is a point whose pseudocharacter is $\sup A$ and whose character and tightness are both $\max\operatorname{pcf}(A)$. Passing to the whole space, we find that its pseudocharacter is always $\delta$, while its character and tightness exceed $\delta$ precisely when the pcf condition fails.

Throughout this section, $\mu<\delta$ are infinite cardinals, $\operatorname{cf}(\delta)=\omega$, and we assume that $X=X(\mu,\delta)$ has the presentation $X(\mu,\kappa)$ where $\kappa$ is strictly increasing. In this context, we define

where we set $\sup\varnothing=0$. Our next goal is the following computation of the character, pseudocharacter, and tightness of $X=X(\mu,\delta)$.

The term $\delta$ in the second clause is necessary in general. For example, if $\delta=\mu^{+\omega}$, then $S_{\mu,\delta}=\varnothing$, while all three global invariants equal $\delta$. The next lemma computes all three invariants at a nonisolated point.

and

Then $A_x$ is a nonempty countable set of regular cardinals that is bounded below $\delta$, and

while

Proof. Since $x$ is nonisolated, we know $I_x \ne\varnothing$, and $\sup A_x < \delta$ by (1). For each $n \in I_x$ choose a cofinal $D_n \subseteq x(n)$ of order type $\operatorname{cf}(x(n))$ consisting of ordinals of cofinality $\mu^+$. Indeed, for $\alpha< x(n)$ the first ordinal of cofinality greater than $\mu$ above $\alpha$ is $\alpha+\mu^+$, and nonisolation puts it below $x(n)$.

Put $D=\prod_{n\in I_x}D_n$, viewing each member $d\in D$ as a point of $X$ by setting $d(n)=x(n)$ for every $n\notin I_x$. For $d\in D$, let $U_d$ consist of those $y\in X$ that agree with $x$ off $I_x$ and satisfy $d(n)<y(n)\le x(n)$ for every $n\in I_x$. Since $x(n)$ is isolated in $T_n$ off $I_x$ and each $D_n$ is cofinal in $x(n)$ on $I_x$, the boxes $U_d$ form a neighborhood base at $x$. A family of these boxes isolates $x$ exactly when its lower bounds are cofinal in each $x(n)$ for $n\in I_x$. If they are bounded in one coordinate, replacing $x(n)$ by a member of $D_n$ above the bound gives a second point in the intersection. Since $I_x$ is countable, $\psi(x,X)=\sup A_x$.

For $E\subseteq D$, the same boxes give $x\in\operatorname{cl}_X E$ exactly when $E$ is pointwise cofinal in $D$. Hence

Identify each $D_n$ with its order type. The functions constant on coordinates of equal cofinality form a cofinal copy of $\prod A_x$ in $D$, since every countable subset of $\theta\in A_x$ is bounded in $\theta$. Thus $\operatorname{cof}(D,\le)=\max\operatorname{pcf}(A_x)$ by (3). □

Proof of Theorem 4.1. Given a nonempty countable $A\subseteq(\mu,\delta)\cap\operatorname{Reg}$ with $\sup A<\delta$, we first find a point $x\in X$ with $A_x=A$. Fix an $\omega$-sequence $\langle\theta_i:i<\omega\rangle$ with range $A$, allowing repetitions. In our fixed increasing presentation $X=X(\mu,\overline{\kappa})$, let

Set $x(n)=\mu^+$ for $n<n^*$ and

where the product is ordinal multiplication. The ordinal $\mu^+\cdot\theta_i$ has cardinality and cofinality $\theta_i$, so it lies below $\kappa_{n^*+i}$, and $x$ is in $X$ because the cofinalities of its values are at most $\sup A$, hence bounded below $\delta$.

For each $i<\omega$, the ordinals $\mu^+\cdot(\xi+1)$ for $\xi<\theta_i$ have cofinality $\mu^+$ and form a sequence in $T_{n^*+i}$ increasing to $x(n^*+i)$, so $x(n^*+i)$ is nonisolated in $T_{n^*+i}$ and hence $n^*+i\in I_x$. Since $x(n)=\mu^+$ is isolated in $T_n$ for $n<n^*$, we have $I_x=\{n^*+i:i<\omega\}$. Hence

By Lemma 4.2, every nonisolated point of $X$ has pseudocharacter below $\delta$. Taking $A=\{\theta\}$ gives a point of pseudocharacter $\theta$ for every regular $\theta\in(\mu,\delta)$. Since these cardinals are cofinal in $\delta$, we have $\psi(X)=\delta$. Taking suprema of the pointwise character and tightness gives

In particular, this supremum is at least $\delta$.

We can now compute (10) in terms of pseudopowers. Let $A$ occur in that supremum and let $\lambda=\max\operatorname{pcf}(A)$. Choose an ultrafilter $U$ on $A$ such that $\operatorname{cf}(\prod A/U)=\lambda$. If $U$ is principal, then $\lambda\in A$, so $\lambda<\delta$. Otherwise, let $\nu$ be the least ordinal for which $A\cap\nu\in U$, and put $B=A\cap\nu$. If $\nu=\beta+1$, then nonprincipality gives $A\cap\beta=B\setminus\{\beta\}\in U$, contrary to the choice of $\nu$. Thus $\nu$ is a limit ordinal, and minimality gives $\sup B=\nu$. Since $B$ is a countable set of cardinals, $\nu$ is a singular cardinal of countable cofinality, and $\mu<\nu<\delta$. The restriction of $U$ to $B$ contains every cobounded subset of $B$. Its dual ideal $J$ therefore extends the bounded ideal, and

Thus every term in (10) is at most $\max\{\delta,H_{\mu,\delta}\}$.

For the reverse inequality, let $\nu\in S_{\mu,\delta}$, and let a regular $\lambda$ be represented at $\nu$ by $\operatorname{tcf}(\prod B/J)=\lambda$. Since $J$ extends the bounded ideal, we can assume $\mu<\min(B)$, and therefore $B$ is a countable subset of $\operatorname{Reg}\cap(\mu,\delta)$ bounded by $\nu<\delta$, and $\lambda\in\operatorname{pcf}(B)$. Thus,

Taking the supremum over the representations of each $\nu$, and then over $\nu\in S_{\mu,\delta}$, gives the lower bound $H_{\mu,\delta}$. Together with the lower bound $\delta$ already obtained, this gives $\chi(X)=t(X)=\max\{\delta,H_{\mu,\delta}\}$. If $S_{\mu,\delta}$ is cofinal in $\delta$, the inequalities $\operatorname{pp}(\nu)\ge\nu^{+}$ give $H_{\mu,\delta}\ge\delta$, so the term $\delta$ may be omitted. □

Proof of Corollary 1.3. By Theorem 1.1, $X$ is nonnormal precisely when some nonempty countable $A\subseteq(\mu,\delta)\cap\operatorname{Reg}$ with $\sup A<\delta$ satisfies $\max\operatorname{pcf}(A)\ge\delta$. Since $\max\operatorname{pcf}(A)$ is regular and $\delta$ is singular, this inequality is strict. It follows from (10) that

Theorem 4.1 gives $\psi(X)=\delta$ and $\chi(X)=t(X)\ge\delta$, so the four conditions are equivalent. □

Remark 4.3. If $X$ is nonnormal, choose $A$ as in the preceding proof. The construction in the proof of Theorem 4.1 gives a point $x\in X$ with $A_x=A$, so by Lemma 4.2,

Conversely, any such point forces $\chi(X)>\delta$, so $X$ is nonnormal.

Department of Mathematics, Ohio University, Athens, OH 45701 USA

Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105 Israel

URL: https://www.math.bgu.ac.il/~kojman/