[Paper Review] Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. III

Paolo Lipparini|ArXiv.org|Apr 23, 2008

Advanced Topology and Set Theory15 references3 citations

TL;DR

This paper establishes a deep connection between ultrafilter regularity and compactness in topological spaces, particularly for singular cardinals. It proves that a topological space $ S_{\lambda}(\lambda) $ is $ D $-compact if and only if the ultrafilter $ D $ is not $ (\lambda,\lambda) $-regular, generalizing earlier results to singular cardinals and linking model-theoretic ultrafilter properties to topological compactness via product and ultraproduct constructions.

ABSTRACT

We generalize the results from "P. Lipparini, Productive $[λ,μ]$-compactness and regular ultrafilters, Topology Proceedings, 21 (1996), 161--171"; in particular the present results apply to singular cardinals, too.

Motivation & Objective

• To generalize results on ultrafilter regularity and topological compactness to singular cardinals, extending prior work on regular cardinals.
• To establish a precise equivalence between $ D $-compactness of specific topological spaces and the non-regularity of ultrafilters.
• To investigate the productively $[\lambda,\mu]$-compactness of topological spaces and its dependence on ultrafilter regularity.
• To characterize when a topological space is $[\kappa_i,\kappa_i]$-compact based on ultrafilter properties, particularly for families of cardinals $ (\kappa_i)_{i\in I} $.
• To introduce and analyze the Frechet disjoint union construction as a tool for preserving topological properties like $ D $-compactness and normality under ultrafilter limits.

Proposed method

• The paper uses the space $ S_\lambda(\mu) $, identified with the set of subsets of $ \mu $ of size less than $ \lambda $, endowed with a topology generated by sets $ \{x \in S_\lambda(\mu) \mid \alpha \in x\} $ and their complements.
• It establishes that $ S_\lambda(\lambda) $ is $ D $-compact if and only if $ D $ is not $ (\lambda,\lambda) $-regular, using ultrafilter convergence and the definition of $ D $-convergence to $ x \in S_\lambda(\lambda) $.
• The proof relies on the fact that $ D $-convergence to $ x $ requires that $ \{i \mid \alpha \in f(i)\} \in D $ for all $ \alpha \in x $, and that $ x $ must have size less than $ \lambda $.
• The Frechet disjoint union $ X = \{x\} \dot{\cup} \dot{\bigcup}_{i\in I} X_i $ is constructed to preserve $ D $-compactness and topological properties like regularity and normality.
• The equivalence between productively $[\lambda,\mu]$-compactness and $ (\kappa_i,\kappa_i) $-regularity of ultrafilters is proven via contradiction, using the existence of a $ (\lambda,\mu) $-regular ultrafilter not $ (\kappa_i,\kappa_i) $-regular for any $ i \in I $.
• The paper applies known results from [C2, Theorem 1.7] and [L2, Proposition 1] to link $ D $-compactness of products to ultrafilter regularity, especially in the context of singular cardinals.

Experimental results

Research questions

• RQ1 For a singular cardinal $ \lambda $, when is the space $ X = \lambda^+ \cup \operatorname{cf}\lambda $ (with order topology) $ D $-compact for an ultrafilter $ D $?
• RQ2 What is the precise relationship between $ D $-compactness of $ S_\lambda(\lambda) $ and the $ (\lambda,\lambda) $-regularity of $ D $?
• RQ3 Under what conditions is a productively $[\lambda,\mu]$-compact space necessarily $[\kappa_i,\kappa_i]$-compact for some $ i \in I $?
• RQ4 How does the Frechet disjoint union construction preserve $ D $-compactness and topological properties like normality and regularity?
• RQ5 When are the conditions of productively $[\lambda,\mu]$-compactness equivalent across different classes of topological spaces (e.g., Hausdorff, normal, Tychonoff groups with clopen bases)?

Key findings

• The space $ S_\lambda(\lambda) $ is $ D $-compact if and only if the ultrafilter $ D $ is not $ (\lambda,\lambda) $-regular, establishing a precise duality between topological compactness and ultrafilter regularity.
• For a singular cardinal $ \lambda $, an ultrafilter is $ (\lambda,\lambda) $-regular if and only if it is either $ (\operatorname{cf}\lambda,\operatorname{cf}\lambda) $-regular or $ (\lambda^+,\lambda^+) $-regular, generalizing earlier results to singular cardinals.
• The topological space $ X = \lambda^+ \cup \operatorname{cf}\lambda $ (with order topology) is $ D $-compact if and only if $ D $ is not $ (\lambda,\lambda) $-regular, and thus $ X $ is not productively $[\lambda,\lambda]$-compact.
• A productively $[\lambda,\mu]$-compact topological space is $[\kappa_i,\kappa_i]$-compact for some $ i \in I $ if and only if every $ (\lambda,\mu) $-regular ultrafilter is $ (\kappa_i,\kappa_i) $-regular for some $ i \in I $, establishing a complete equivalence.
• The Frechet disjoint union of $ D $-compact spaces is $ D $-compact, and if each component space has a base of clopen sets, so does the union, preserving key topological properties.
• When each $ \kappa_i $ is regular or a singular cardinal of cofinality $ \omega $, the conditions in Theorem 6 remain equivalent, including the requirement that $ D $-compactness implies $[\kappa_i,\kappa_i]$-compactness for some $ i \in I $.