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.

Cardinal Functions I

BY W. W. COMFORT I am grateful to Peter Freyd for his generous introductory comments, and I am grateful also to the [Selection] Committee for extending the invitation to speak to you. Last month a colleague with whom I attempted to discuss today's remarks suggested drily that as half of the team responsible for [CN2] I have probably already said more than enough about ultrafllters, and that if I insist on pursuing the matter further today I could do so most gracefully and efficiently simply by offering a complimentary copy of [CN2] to each of you. Eschewing that advice I shall in the hour allotted to me attempt to achieve the following three goals. (A) To acquaint you with what I think are some of the most basic, fundamental facts about ultrafllters on a discrete topological space; this material is sufficiently simple and elegant that it can be absorbed comfortably into a first-year graduate course in general topology. (B) To give some partial results, less definitive and less conclusive than the optimal theorems available, concerning the existence of particular ultrafllters with special properties; I hope that the results chosen in this connection have the complementary virtues that they are sufficiently powerful to handle most of the situations treated by the more powerful results which we shall ignore, and that their proofs are significantly simpler than those of the more general results. (C) To record some results about ultrafllters which came to my attention after the publication of [CN2]; I have chosen today to emphasize three relatively new results which are not formally concerned with ultrafllters and which indeed make no mention of ultrafllters in their statements, but which nevertheless have been given proofs in which ultrafllters play an important catalytic role. My hope is that (even) those of you not professionally inclined toward topology or set theory will find something potentially useful, or amusing, among the basic results given in (A). The theorems selected for inclusion in (B) are given not only because of the intrinsic beauty and elegance of their proofs, but also because they serve to indicate the principal sorts of questions

/pdf/ultrafilters-some-old-and-some-new-results-pni16310sa.pdf

Ultrafilters: Some old and some new results

A path of a graph is called maximal if it is not a proper subpath of any other path of the graph. The path spectrum of a graph G, denoted by ps(G), is the set of lengths of all maximal paths in the graph. A set S of positive integers is called a path spectrum if there is a connected graph G such that ps(G) = S. Jacobson et al. showed that all sets of positive integers with cardinality of 1 or 2 are path spectrum sets. Their results raised the question of whether all sets of positive integers are path spectra. We show that, for every positive integer k ≥ 3, there are infinitely many sets of k positive integers which are not path spectra. A set S of positive integers is called an absolute path spectrum if there are infinitely many connected graphs G such that ps(G) = S. We completely characterize absolute path spectra S for |S| ≤ 2. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 329–350, 2008

Non-path spectrum sets

Publisher Summary This chapter discusses initially κ-compact and related spaces. A main reason for studying initial κ and, more generally, interval compactness is that compactness, countable compactness, and the Lindelof property are special cases of one or both of these concepts. Another reason is that the theory of initially κ-compact and related spaces provides a means for answering fundamental questions that arise in other areas of topology. A third reason is that results in this area illustrate the usefulness of the close relationship that exists between the set theory of uncountable cardinals and properties of topological spaces. A space is countably compact if and only if it is initially ω-compact, and it is Lindelof if and only if it is finally ω1-compact. Like compactness, initial κ-compactness is preserved by continuous mappings, perfect pre-images, and closed subsets.

Initially κ-Compact and Related Spaces

where, for a < h, B,~ is a subalgebra of B and a </3 < h implies B~ _ B~, B~ ~: B~ (we identify cardinals with initial ordinals and each ordinal with the set of smaller ordinals). The question how cf(B) does "look like" makes sense, of course, for other types of structures than Boolean algebras; see [2] and [5]. It is proved in [2] that there is a group G s.t. cf(G) is uncountable, one of the arguments being that there is a BA B s.t. cf(B) is uncountable. We shall show that in all cases we are able to handle, cf(B) equals No or N1, but we do not know whether this holds for every BA B. We shall denote the finitary operations on a BA B by + , . , , 0, 1; the infinite ones by Y. and l-[. Let us first remark that of(B) is well-defined: suppose card(B)=h-->No. Let B = {b~ I a < )t} and define, for a < h, B~ to be the subalgebra of B generated by {b~[~< a}. Each B~ is a proper subalgebra of B (for h = No, this holds since a finitely generated BA is finite), and thus, cf(B)-< h = card(B). There are some very simple facts:

Boolean algebras as unions of chains of subalgebras

f(x) sei eine Abbildung von A in A*, d. h. jedem x∈ A ist ein f(x)∈ A* zugeordnet. Beliebig viele solche Abbildungen mogen wesentlich verschieden heisen, wenn es zu endlich vielen verschiedenen f1,...,fk unter ihnen immer mindestens eine Stelle x gibt, wo f 1 (x),...,f k (x) paarweise verschieden sind.

Über zwei Sätze von G. Fichtenholz und L. Kantorovitch

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Über zwei Sätze von G. Fichtenholz und L. Kantorovitch