---
abstract: 'We generalize the results from [@topproc]; in particular the present results apply to singular cardinals, too.'
address: |
    Dipartimento di Matematica\
    Viale della Ricerca Scientifica\
    II Universitaccia Romanaccia (Tor Vergata)\
    I-00133 ROME ITALY 
author:
- Paolo Lipparini
title: 'Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. III.'
---

See [@parti; @KV; @EGT] for definitions and notation.

We shall need the following theorem from [@nd].

\[limit\] If $ \lambda $ is a singular cardinal, then an ultrafilter is $(\lambda ,\lambda )$-regular if and only if it is either $(\operatorname{cf}\lambda ,\operatorname{cf}\lambda )$-regular or $(\lambda^+ ,\lambda^+ )$-regular.

\[cor\] Suppose that $ \lambda $ is a singular cardinal, and consider the topological space $X$ obtained by forming the disjoint union of the topological spaces $ \lambda ^+$ and $\operatorname{cf}\lambda $, both endowed with the order topology.

Then, for every ultrafilter $D$, the space $X$ is $D$-compact if and only if $D$ is not $( \lambda, \lambda )$-regular.

Thus, $X$ is productively $[ \lambda', \mu']$-compact if and only if there exists a $( \lambda', \mu' )$-regular not $( \lambda, \lambda )$-regular ultrafilter. In particular, $X$ is not productively $[ \lambda, \lambda ]$-compact.

By Theorem \[limit\], $D$ is not $( \lambda, \lambda )$-regular if and only if it is neither $(\operatorname{cf}\lambda ,\operatorname{cf}\lambda )$-regular nor $(\lambda^+ ,\lambda^+ )$-regular.

Hence, by [@topproc Proposition 1], and since both $ \lambda ^+$ and $\operatorname{cf}\lambda $ are regular cardinals, $D$ is not $( \lambda, \lambda )$-regular if and only if both $ \lambda ^+$ and $\operatorname{cf}\lambda $ are $D$-compact. This is clearly equivalent to $X$ being $D$-compact.

The last statement is immediate from [@C Theorem 1.7], also stated in [@topproc Theorem 2].

Let $\mathbf{2}= \{0,1 \} $ denote the two-elements topological space with the discrete topology. If $ \lambda \leq \mu$ are cardinals, let $\mathbf{2}^ \mu $ be the Tychonoff product of $ \mu $-many copies of $\mathbf{2}$, and let $\mathbf{2}^ \mu_ \lambda  $ denote the subset of $\mathbf{2}^ \mu $ consisting of all those functions $h: \mu \to  \mathbf{2}$ such that $ \left| \{ \alpha \in \mu| h( \alpha )=1 \} \right| < \lambda $.

In passing, let us mention that, when $ \mu= \aleph_\omega$, the space $\mathbf{2}^ \mu_ \mu  $ provides an example of a linearly Lindelöf not Lindelöf space. See [@AB Example 4.1]. Compare also [@St Example 4.2].

Notice that $\mathbf{2}^ \mu_ \lambda  $ is a Tychonoff topological group with a base of clopen sets.

Set theoretically, $\mathbf{2}^ \mu_ \lambda  $ is in a one to one correspondence (via characteristic functions) with $S_\lambda(\mu)$, the set of all subsets of $ \mu$ of cardinality $ < \lambda $. Since many properties of ultrafilters are defined in terms of $S_\lambda(\mu)$, for sake of convenience, in what follows we shall deal with $S_\lambda(\mu)$, rather than $\mathbf{2}^ \mu_ \lambda  $. Henceforth, we shall deal with the topology induced on $S_\lambda(\mu)$ by the above correspondence.

In detail, $S_\lambda(\mu)$ is endowed with the smallest topology containing, as open sets, all sets of the form $X_\alpha = \{x\in S_\lambda(\mu) | \alpha\in x\}$ ($\alpha$ varying in $\mu$), as well as their complements. Thus, a base for the topology consists of all finite intersections of the above sets; that is, the elements of the base are the sets $\{x\in S_\lambda(\mu) | \alpha_1\in x,  \alpha_2\in x , \dots, \alpha_n\in x , \beta_1\not\in x,  \beta_2\not\in x , \dots, \beta_m\not\in x \}$, with $n,m$ varying in $\omega$ and $\alpha_1, \dots, \alpha_n, \beta_1 , \dots, \beta_m
$ varying $\mu$.

Notice that this topology is finer than the topology on $S_\lambda(\mu)$ used in [@topproc].

With the above topology, $S_\lambda(\mu)$ and $\mathbf{2}^ \mu_ \lambda  $ are homeomorphic, thus $S_\lambda(\mu)$ can be given the structure of a Tychonoff topological group.

Notice that if $ \lambda \leq \mu$ then $S_\lambda(\mu)$ is not $[\lambda ,\lambda ]$-compact. Indeed, for $ \alpha \in \mu$, let $Y_\alpha = \{ x \in S_\lambda(\mu)| \alpha \not\in x \} $. If $Z \subseteq \mu$ and $|Z|= \lambda $ then $(Y_\alpha )_{\alpha\in Z}$ is an open cover of $S_\lambda(\mu)$ by $\lambda $-many sets, $<\lambda $ of which never cover $S_\lambda(\mu)$.

\[dsll\] For every ultrafilter $D$ and every cardinal $\lambda$, the topological space $S_\lambda(\lambda)$ is $D$-compact if and only if $D$ is not ($\lambda,\lambda$)-regular.

Suppose that $D$ is an ultrafilter over $I$ and that $S_\lambda(\lambda)$ is $D$-compact. For every $f:I\to S_\lambda(\lambda)$ there exists $x\in S_\lambda(\lambda)$ such that $f(i)_{i\in I}$ $D$-converges to $x$. If $\alpha\in\lambda  $ and $\{i\in I|\alpha\in f(i)\}\in D$ then $\alpha\in x $, since otherwise $Y=\{z\in S_\lambda(\lambda) | \alpha\not\in z\}$ is an open set containing $x$, and $\{i\in I|f(i)\in Y\}= \{i\in I|\alpha\not\in f(i)\}\not\in D$, contradicting $D$-convergence.

Whence, $\{\alpha\in\lambda  |\{i\in I|\alpha\in f(i)\}\in D\}\subseteq x\in S_\lambda(\lambda)$, and thus $x$ has cardinality $<\lambda$; that is, $f$ does not witness ($\lambda,\lambda$)-regularity of $D$. Since $f$ has been chosen arbitrarily, $D$ is not ($\lambda,\lambda$)-regular.

Conversely, suppose that $D$ over $I$ is not ($\lambda,\lambda$)-regular, and let $f:I\to S_\lambda(\lambda)$. Then $x=\{\alpha\in\lambda  |\{i\in I|\alpha\in f(i)\}\in D\}$ has cardinality $<\lambda$ and hence is in $S_\lambda(\lambda)$. We show that $f$ $D$-converges to $x$. Indeed, let $Y$ be a neighborhood of $x$: we have to show that $\{i\in I|f(i)\in Y\}\in D$. Without loss of generality, we can suppose that $Y$ is an element of the base of $S_\lambda(\lambda)$, that is, $Y$ has the form $\{z\in S_\lambda(\lambda) | \alpha_1\in z,  \alpha_2\in z , \dots, \alpha_n\in z , \beta_1\not\in z,  \beta_2\not\in z , \dots, \beta_m\not\in z \}$. Since $D$ is closed under finite intersections, then $\{i\in I|f(i)\in Y\}\in D$ if and only if $\{i\in I|\alpha_1\in f(i)\}\in D$ and $\{i\in I|\alpha_2\in f(i)\}\in D$ and… and $\{i\in I|\alpha_n\in f(i)\}\in D$ and $\{i\in I|\beta_1\not\in f(i)\}\in D$ and… and $\{i\in I|\beta_m\not\in f(i)\}\in D$. But all the above sets are actualy in $D$, by the definition of $x$ and since $x\in Y$ and $D$ is an ultrafilter; thus $f$ $D$-converges to $x$.

Since $f$ was arbitrary, every $f:I\to S_\lambda(\lambda)$ $D$-converges, and thus $S_\lambda(\lambda)$ is $D$-compact.

\[psll\] The space $S_\lambda(\lambda)$ is productively $[\lambda',\mu']$-compact if and only if there exists a $(\lambda',\mu')$-regular not-$(\lambda,\lambda)$-regular ultrafilter.

Immediate from Proposition \[dsll\] and [@C Theorem 1.7].

In the statements of the next theorems the word “productively”, when included within parentheses, can be equivalently inserted or omitted.

\[topprocsing\] For all infinite cardinals $ \lambda  $, $ \mu$, $ \kappa $, the following are equivalent:

\(i) Every productively $[ \lambda, \mu]$-compact topological space is (productively) $[ \kappa , \kappa ]$-compact.

\(ii) Every productively $[ \lambda, \mu]$-compact family of topological spaces is productively $[ \kappa , \kappa ]$-compact.

\(iii) Every $( \lambda, \mu)$-regular ultrafilter is $( \kappa , \kappa )$-regular.

\(iv) Every productively $[ \lambda, \mu]$-compact Hausdorff normal topological space with a base of clopen sets is productively $[ \kappa , \kappa ]$-compact.

\(v) Every productively $[ \lambda, \mu]$-compact Tychonoff topological group with a base of clopen sets is (productively) $[ \kappa , \kappa ]$-compact.

If $ \kappa $ is regular, then the preceding conditions are also equivalent to:

\(vi) Every productively $[ \lambda, \mu]$-compact Hausdorff normal topological space with a base of clopen sets is $[ \kappa , \kappa ]$-compact.

Let us denote by (i)$_{\text p}$ Condition (i) when the second occurrence of the word “productively” is included, and simply by (i) when it is omitted. Similarly, for condition (v).

The equivalence of (i)-(iii) has been proved in [@topproc Theorem 1], where it has also been proved that, for $ \kappa $ regular, they are equivalent to (vi).

Since (ii) $ \Rightarrow $ (i)$_{\mathrm p}$ $ \Rightarrow $ (i) are trivial, we get that (i), (ii), (iii), (i)$_{\mathrm p}$ are all equivalent, and equivalent to (vi) for $\kappa $ regular.

\(ii) $\Rightarrow $ (iv) and (ii) $\Rightarrow $ (v)$_{\mathrm p}$ $ \Rightarrow $ (v) are trivial.

If (iii) fails, then there is a $( \lambda, \mu)$-regular ultrafilter which is not $( \kappa , \kappa )$-regular, thus, for $\kappa $ singular, the space $X$ of Corollary \[cor\] is productively $[ \lambda, \mu]$-compact. For $\kappa $ regular, take $X= \kappa $ with the order topology (see [@topproc]). $X$ is Hausdorff, normal, with a base of clopen sets, but not productively $[ \kappa , \kappa ]$-compact, again by Corollary \[cor\], thus (iv) fails. We have proved (iv) $\Rightarrow $ (iii).

\(v) $\Rightarrow $ (iii) is similar, using Corollary \[psll\], since $S_\kappa (\kappa )$ is not $[\kappa ,\kappa ]$-compact.

\[topproc2sing\] For all infinite cardinals $ \lambda  $, $ \mu$, and for any family $ (\kappa_i)_{i \in I} $ of infinite cardinals, the following are equivalent:

\(i) Every productively $[ \lambda, \mu]$-compact topological space is (productively) $[ \kappa_i , \kappa_i ]$-compact for some $i \in I$.

\(ii) Every productively $[ \lambda, \mu]$-compact family of topological spaces is productively $[ \kappa_i , \kappa_i ]$-compact for some $i \in I$.

\(iii) Every $( \lambda, \mu)$-regular ultrafilter is $( \kappa_i , \kappa_i )$-regular for some $i \in I$.

\(iv) Every productively $[ \lambda, \mu]$-compact Hausdorff normal topological space with a base of clopen sets is productively $[ \kappa_i , \kappa_i ]$-compact for some $i \in I$.

\(v) Every productively $[ \lambda, \mu]$-compact Tychonoff topological group with a base of clopen sets is (productively) $[ \kappa_i , \kappa_i ]$-compact for some $i \in I$.

If every $ \kappa_i $ is regular, then the preceding conditions are also equivalent to:

\(vi) Every productively $[ \lambda, \mu]$-compact Hausdorff normal topological space with a base of clopen sets is $[ \kappa_i , \kappa_i ]$-compact for some $i \in I$.

The equivalence of (i)-(iii) has been proved in [@topproc Theorem 3], thus, arguing as in the proof of Theorem \[topprocsing\], we get that (i), (ii), (iii), (i)$_{\mathrm p}$ are all equivalent.

\(ii) $\Rightarrow $ (iv) $\Rightarrow $ (vi) and (ii) $\Rightarrow $ (v)$_{\mathrm p}$ $ \Rightarrow $ (v) are trivial.

If (iii) fails, then there is a $( \lambda, \mu)$-regular ultrafilter $D$ which for no $i \in I$ is $( \kappa_i , \kappa_i )$-regular. By Proposition \[dsll\], for every $i \in I$ the topological space $S_{\kappa_i} (\kappa_i )$ is $D$-compact. Hence $X = \prod_{i \in I} S_{\kappa_i} (\kappa_i )$ is $D$-compact, thus productively $[ \lambda, \mu]$-compact, by [@C Theorem 1.7]. However, $X$ is a Tychonoff topological group with a base of clopen sets which for no $i \in I$ is $[ \kappa_i , \kappa_i ]$-compact, thus (v) fails. We have proved (v) $\Rightarrow $ (iii).

The proofs of (iv) $\Rightarrow $ (iii) and (vi) $\Rightarrow $ (iii) are similar, using the next proposition. If (iii) fails, then there is a $( \lambda, \mu)$-regular ultrafilter $D$ which for no $i \in I$ is $( \kappa_i , \kappa_i )$-regular. By the proof of Theorem \[topprocsing\], for every $i \in I$ we have a $D$-compact topological space $X_i $ which falsify \[topprocsing\](iv), resp., \[topprocsing\](vi). Then the space $X = \{x\}  {\dot{\cup}}{\dot{\bigcup}}_{i \in I} X_i $ we shall construct in the next definition is $D$-compact, thus productively $[ \lambda, \mu]$-compact, by [@C Theorem 1.7], and makes (iv), resp., (vi), fail.

\[fr\] Given a family $(X_i) _{i \in I} $ of topological spaces, construct their *Frechet disjoint union* $X = \{x\}  {\dot{\cup}}{\dot{\bigcup}}_{i \in I} X_i $ as follows.

Set theoretically, $X$ is the union of (disjoint copies) of the $X_i$’s, plus a new element $x$ which belongs to no $X_i$. The topology on $X$ is the smallest topology which contains each open set of each $X_i$, and which contains $\{x\}  {\dot{\cup}}{\dot{\bigcup}}_{i \in E} X_i $, for every $E \subseteq I$ such that $I \setminus E$ is finite.

\[disj\] If $(X_i) _{i \in I} $ is a family of topological spaces, then their Frechet disjoint union $X = \{x\}  {\dot{\cup}}{\dot{\bigcup}}_{i \in I} X_i $ is $T_0$, $T_1$, Hausdorff, regular, normal, $D$-compact (for a given ultrafilter $D$), $[\lambda,\mu]$-compact (for given infinite cardinals $\lambda $ and $\mu$), has a base of clopen sets if and only if so is (has) each $X_i$.

Straightforward. We shall comment only on regularity, normality and $D$-compactness.

For regularity and normality, just observe that if $C$ is closed in $X$ and $C$ has nonempty intersection with infinitely many $X_i$’s, then $x \in C$.

As for $D$-compactness, suppose $D$ is over $J$ and that each $X_i$ is $D$-compact. Let $(y_j) _{j \in J} $ be a sequence of elements of $X$. If $ \{j \in J| y_j \in \bigcup _{i \in F} X_i \} \not\in D $ holds for every $F \subseteq I$, then $(y_j) _{j \in J} $ $D$-converges to $x$. Otherwise, since $D$ is an ultrafilter, hence $\omega$-complete, there exists some $i \in I$ such that $ \{j \in J| y_j \in  X_i \} \in D $. But then $(y_j) _{j \in J} $ $D$-converges to some point of $X_i$, since $X_i$ is supposed to be $D$-compact.

When $ \kappa $ is singular of cofinality $ \omega $, Condition (vi) in Theorem \[topprocsing\] is equivalent to the other conditions. When each $ \kappa_i $ is either a regular cardinal, or a singular cardinal of cofinality $ \omega $, then Condition (vi) in Theorem \[topproc2sing\]. is equivalent to the other conditions. Proofs shall be given elsewhere.

A. V. Arhangel’skii, R. Z. Buzyakova, [*On linearly Lindelöf and strongly discretely Lindelöf spaces.*]{} Proc. Amer. Math. Soc. 127 (1999), no. 8, 2449–2458.

X. Caicedo, [*On productive $[\kappa,\lambda]$-compactness, or the Abstract Compactness Theorem revisited*]{}, manuscript (1995).

X. Caicedo, [*The Abstract Compactness Theorem Revisited*]{}, in [*Logic and Foundations of Mathematics (A. Cantini et al. editors),*]{} Kluwer Academic Publishers (1999), 131–141.

K. P. Hart, J. Nagata, J. E. Vaughan (editors), [*Encyclopedia of General Topology*]{}, Amsterdam (2003).

A. Kanamori and M. Magidor, [*The evolution of large cardinal axioms in Set Theory*]{}, in: [*Higher Set Theory*]{}, edited by G. H. Müller and D. S. Scott, 99–275, Berlin (1978).

K. Kunen and J. E. Vaughan (editors), [*Handbook of Set Theoretical Topology*]{}, Amsterdam (1984).

P. Lipparini, [*Ultrafilter translations, I: $(\lambda,\lambda )$-compactness of logics with a cardinality quantifier*]{}, Arch. Math. Logic [**35**]{}, 63–87 (1996).

P. Lipparini, [*Productive $[\lambda,\mu ]$-compactness and regular ultrafilters*]{}, Topology Proceedings **21** (1996), 161–171.

P. Lipparini, [*Compact factors in finally compact products of topological spaces*]{}, Topology and its Applications, **153** (2006), 1365–1382.

P. Lipparini, [*Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I*]{}, arXiv:0803.3498; [*II.*]{}:0804.1445 (2008).

P. Lipparini, [*Decomposable ultrafilters and possible cofinalities*]{}, in print on Notre Dame Journal of Formal Logic.

R. M. Stephenson, [*Initially $\kappa $-compact and related spaces*]{}, Chapter 13 in [@KV], 603–632.
