Topic
Cardinal function
About: Cardinal function is a research topic. Over the lifetime, 275 publications have been published within this topic receiving 4367 citations.
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TL;DR: In this article, the properties of n-free and omega-free boolean algebras have been investigated and connections to hypergraph theory and cardinal invariants on them have been made.
Abstract: We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -independent subset; and i_omega, the minimum size of an omega-independent subset, are introduced and investigated. The values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC. Ideal-independence is also considered, and it is shown that the cardinal function p <= s_mm for infinite boolean algebras. We also define and consider moderately generated boolean algebras; that is, those boolean algebras that have a generating set consisting of elements that split finitely many elements of the boolean algebra.
1,502 citations
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TL;DR: Cardinal functions as mentioned in this paper extend important topological properties such as countable base, separable, and first countable to higher cardinality, allowing one to formulate, generalize, and prove results of a particular type in a systematic and elegant manner.
Abstract: Publisher Summary This chapter discusses cardinal functions. Cardinal functions extend important topological properties as countable base, separable, and first countable to higher cardinality. Cardinal functions allow one to formulate, generalize, and prove results of a particular type in a systematic and elegant manner. The cardinal functions also allow one to make precise quantitative comparisons between certain topological properties. Prerequisite for work in cardinal functions is knowledge of cardinal and ordinal numbers and transfinite constructions. The chapter also describes the cardinal functions on the two most important classes of abstract topological spaces, namely, compact spaces and metrizable spaces. It also describes cardinal functions that are used to obtain bounds on the cardinality of a space. An infinite cardinal that is not regular is said to be singular. Theorems from combinatorial set theory play an important role in cardinal functions.
306 citations
1 Jan 1978
203 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 9 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 2 |
| 2017 | 5 |
| 2016 | 5 |