Cardinal Invariants Research Articles

Cardinal invariants concerning bounded families of extendable and almost continuous functions

In this paper we introduce and examine a cardinal invariant A b \operatorname {A}_{{b}} closely connected to the addition of bounded functions from R \mathbb {R} to R \mathbb {R} . It is analogous to the invariant A \operatorname {A} defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.

On cardinal invariants for CCC 𝜎-ideals

We show several results about cardinal invariants for σ \sigma -ideals of the reals. In particular we show that for every CCC σ \sigma -ideal on the real line p ≤ cof ⁡ ( J ) p \le \operatorname {cof}(J) .

Cardinal invariants of ultraproducts of Boolean algebras

Cardinal invariants of ultraproducts of Boolean algebras

Sticks and clubs

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω 1 together with ¬CH and Martin's axiom for countable p.o.-sets.

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.

Algebraic properties of the class of Sierpiński-Zygmund functions

Sums, products and compositions with Sierpiński-Zygmund functions are investigated. Moreover, cardinal invariants connected with those operations are defined and studied.

Cardinal invariants of monotonically normal spaces

The basic cardinal invariants of monotonically normal spaces are determined. The gap between cellularity and density is investigated via calibres.

Cardinal invariants about shrinkability of unbounded sets

In our previous paper (Eda et al., to appear), we introduced a cardinal invariant b * and studied some properties of the cardinal b *. In the present paper we define new cardinal invariants which are related to Cichofi's diagram and generalize the notion of b *. We investigate the relations between them and other cardinals which appear in Cichoń's diagram.

Monotone normality

The article surveys recent developments on monotone normality in the context of classical theory. Particular results concern protometrisable, elastic and well ordered ( F) spaces, cardinal invariants and compactifications, Borges normal spaces and the extension of continuous functions, Gartside's p-products, topological groups and the structuring mechanism introduced by A.W. Roscoe and the author. A major influence is P.M. Gartside's thesis.

Adding one random real

AbstractWe study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

Rectifiable spaces

The notion of rectifiable space is a generalization of the notion of a topological group. In the article it is shown that a rectifiable first countable T 0 space is metrizable; different cardinal invariants are equal for rectifiable spaces.

C(X) in the weak topology

Some relations between cardinal invariants ofX andC(X) are established in the weak topology, whereC(X) is the space of continuous real-valued functions onX in the compact-open topology.

Cardinal invariants above the continuum

We prove some consistency results about b (λ) and δ(λ), which are natural generalisations of the cardinal invariants of the continuum b and d . We also define invariants b cl(λ) and δ cl ( λ), and prove that almost always b (λ) = b cl(λ) and d (λ) = d cl(λ).

A special class of almost disjoint families

AbstractThe collection of branches (maximal linearly ordered sets of nodes) of the tree <ωω (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal — for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and is the minimum cardinality of a maximal off-branch family.Results concerning include: (in ZFC) , and (consistent with ZFC) is not equal to any of the standard small cardinal invariants or = 2ω. Most of these consistency results use standard forcing notions—for example, in the Cohen model.Many interesting open questions remain, though—for example, whether .

Combinatorial properties of classical forcing notions

We investigate the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum. We show: 1. (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω 1; 2. (2) Laver and Mathias forcing collapse the dominating number to ω 1, and thus two Laver or Mathias reals added iteratively always force CH; 3. (3) Miller's rational perfect set forcing preserves the axiom MA(σ-centered).

Hechler reals

AbstractWe define a σ-ideal on the set of functions ωω with the property that a real x ∈ ωω is a Hechler real over V if and only if x omits all Borel sets in . In fact we define a topology on ωω related to Hechler forcing such that is the family of first category sets in . We study cardinal invariants of the ideal .

Densities of Ultraproducts of Boolean Algebras

AbstractWe answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

The counterparts of some cardinal functions in bitopological spaces. I.

This is the first in a series of papers aimed at defining and studying bitopological counterparts of the principal cardinal invariants in topology. It is devoted to study of analogues of the functions weight, density and cellularity.

H‐Enrichments and Their Homeomorphism Groups II

ABSTRACT. H‐enrichments of topologies are larger (i.e., inclusive) topologies that also have larger homeomorphism groups. The study of H‐enrichments of the Euclidean and rational topologies via generalized Baire category arguments is continued. New results concern lower bounds on the number of H‐enrichments, connectedness, and separation properties of such topologies and certain of their cardinal invariants.

On Topologies Generated by Filters of Relationsa

ABSTRACT. A range of topologies, generated in a “preuniform” manner, is shown to give conditions for a space to be (1) Nagata over a regular infinite cardinal α, and (2) α‐metrizable. Connections with the structuring mechanism introduced by P. J. Collins and A. W. Roscoe are investigated. The metrizability degree of a regular space is shown to be equal to the minimum among cardinals arising as weights of compatible local uniformities, and the reader is asked to characterize topologies admitting monotonic quasi uniformities. Various relevant cardinal invariants are discussed; in particular, comparisons are made involving the transitivity and γ‐degrees.