Sequential Continuity and Submeasurable Cardinals

Submeasurable cardinals are deened in a similar way as measurable cardinals are. Their characterizations are given by means of sequentially continuous pseudonorms (or homomorphisms) on topological groups and of sequentially continuous (or uniformly continuous) functions on Cantor spaces (for that purpose it is proved that if a complete Boolean algebra admits a nonconstant sequentially continuous function, it admits a Maharam submea-sure). It appeared to be convenient to have a hierarchy of large cardinals starting with the rst sequential cardinal. For instance, productivity numbers of certain classes of topological groups or topological vector spaces agree exactly with such large cardinals (see 10] and 7]). For some reasons it seemed to us more natural to deene those cardinals by means of sequentially continuous submeasures in a similar way as measurable cardinals are deened. Let us recall that sequential cardinal is a cardinal such that there exists a sequentially continuous noncontinuous real-valued map on the Cantor space 2. Those cardinals were dealt with in the classical Mazur's paper 13] and in 15]. Mazur showed that the rst sequential cardinal is weakly inaccessible and that every sequentially continuous map on a product of less than-many metrizable separable spaces into a metrizable space is continuous (even a little more general spaces can be used). Noble 15] generalized the class of metrizable separable spaces used in the last mentioned result to a bigger class including rst countable spaces. Keisler and Tarski asked in 12] about relations of the rst sequential cardinal to (real) measurable cardinals. Recall that an uncountable cardinal is measurable (or realmeasurable) if there exists a nonzero-additive measure on all subsets of having zero values at points, with values in 2 (or in R, resp.). The-additivity means that the measure is additive on disjoint families of car-dinalities less than. Clearly, the rst sequential cardinal is not bigger than the rst realmeasurable cardinal (up to now, it is not known whether these two cardinals may diier). Chudnovskij in 3] and in some of his later papers presented some contributions to the above mentioned Keisler{Tarski problem; he showed that the rst sequential cardinal is, in Fremlin's terminology, quasi-measurable (for a deenition, see 5] and the paragraph following our Deenition 1), thus it is bigger than small weakly inaccessible cardinals and, if bigger than 2 ! (which 1 The authors acknowledge the support of the Czech Republic grant 201/97/0216 and the second author also the grants GAUK …