Mathematics Commons^™

Small Combinatorial Cardinal Characteristics And Theorems Of Egorov And Blumberg, Krzysztof Ciesielski

Faculty & Staff Scholarship

We will show that the following set theoretical assumption

• \continuum=\omega₂, the dominating number d equals to \omega₁, and there exists an \omega₁-generated Ramsey ultrafilter on \omega

(which is consistent with ZFC) implies that for an arbitrary sequence f_n:R-->R of uniformly bounded functions there is a subset P of R of cardinality continuum and an infinite subset W of \omega such that {f_n|_P: n in W} is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions f_n are measurable or have the …

Compositions Of Darboux And Connectivity Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

We describe here an example of a Darboux function k from the unit interval I = [0, 1] onto itself such that k is not the composition of any finite collection of connectivity functions from I into I. This answers a question of Ceder [2].

Two Examples Concerning Extendable And Almost Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function f:[0,1]-->[0,1] with a G_\delta graph. This answers a question of Gibson. The second example is that of a connectivity function F:R²-->R with dense graph such that F^-1(0) is contained in a countable union of straight lines. This easily implies the existence of an extendable function f:R-->R with dense graph such that f^-1(0) is countable.

We also give a sufficient condition for a Darboux …

Uniformly Antisymmetric Function With Bounded Range, Krzysztof Ciesielski

Faculty & Staff Scholarship

The goal of this note is to construct a uniformly antisymmetric function f : R → R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski’s survey [5].) A problem of existence of uniformly antisymmetric function f : R → R with finite range remains open.

Some Additive Darboux-Like Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this note we will construct several additive Darboux-like functions f : R → R answering some problems from (an earlier version of) [4]. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions f : R → R whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function f : R → R which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an f does not have the …

Κ-To-1 Darboux-Like Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

We examine the existence of κ-to-1 functions f : R → R in the class of continuous functions, Darboux functions, functions with perfect roads, and functions with the Cantor intermediate value property. In this setting κ denotes a cardinal number (finite or infinite). We also consider different variations on this theme.

Darboux Like Functions That Are Characterizable By Images, Preimages And Associated Sets, Krzysztof Ciesielski

Faculty & Staff Scholarship

For arbitrary families A and B of subsets of R let C(A,B)= {f| f: R-->R and the image f[A] is in B for every A in A} and C^-1 (A,B)= {f| f: R-->R and the inverse image f^-1(B) is in A for every B in B}. A family F of real functions is characterizable by images (preimages) of sets if F=C(A,B) (F=C^-1(A,B), respectively) for some families A and B. …

Symmetrically Continuous Function Which Is Not Countably Continuous, Krzysztof Ciesielski

Faculty & Staff Scholarship

We construct a symmetrically continuous function f : R → R such that for some X ⊂ R of cardinality continuum f|X is of Sierpi´nskiZygmund type. In particular such an f is not countably continuous. This gives an answer to a question of Lee Larson.

Characterizing Derivatives By Preimages Of Sets, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this note we will show that many classes F of real functions f : R → R can be characterized by preimages of sets in a sense that there exist families A and D of subsets of R such that F = C(D, A), where C(D, A) = {f ∈ R R : f −1 (A) ∈ D for every A ∈ A}. In particular, we will show that there exists a Bernstein B ⊂ R such that the family ∆ of all derivatives can be represented as ∆ = C(D, A), where A = S c∈R {(−∞, c),(c, …

Set Theoretic Real Analysis, Krzysztof Ciesielski

Faculty & Staff Scholarship

This article is a survey of the recent results that concern real functions (from Rⁿ into R) and whose solutions or statements involve the use of set theory. The choice of the topics follows the author's personal interest in the subject, and there are probably some important results in this area that did not make to this survey. Most of the results presented here are left without the proofs.

Compositions Of Two Additive Almost Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

In the paper we prove that an additive Darboux function f : R → R can be expressed as a composition of two additive almost continuous (connectivity) functions if and only if either f is almost continuous (connectivity) function or dim(ker(f)) 6= 1. We also show that for every cardinal number λ ≤ 2 ω there exists an additive almost continuous functions with dim(ker(f)) = λ. A question whether every Darboux function f : R → R can be expressed as a composition of two almost continuous functions (see [?] or [?]) remains open.

Cardinal Invariants Concerning Extendable And Peripherally Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

Let F be a family of real functions, F ⊆ R R . In the paper we will examine the following question. For which families F ⊆ R R does there exist g : R → R such that f + g ∈ F for all f ∈ F? More precisely, we will study a cardinal function A(F) defined as the smallest cardinality of a family F ⊆ R R for which there is no such g. We will prove that A(Ext) = A(PR) = c + and A(PC) = 2c , where Ext, PR and PC stand for the …

Uniformly Antisymmetric Functions And K5, Krzysztof Ciesielski

Faculty & Staff Scholarship

A function f from reals to reals (f:R-->R) is a uniformly antisymmetric function if there exists a gage function g:R-->(0,1) such that |f(x-h)-f(x+h)| is greater then or equal to g(x) for every x from R and 0R-->N, (see [K. Ciesielski, L. Larson, Uniformly antisymmetric functions, Real Anal. Exchange 19 (1993-94), 226-235]) while it is unknown whether such function can have a finite or bounded range. It is not difficult to show that there exists a uniformly antisymmetric function with an n-element range if and only if there exists a …

Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous, Krzysztof Ciesielski

Faculty & Staff Scholarship

Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ R R for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F. We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. It is known, that c < A(A) ≤ 2 c [10]. We will generalize this result by showing that the cofinality of A(A) is greater that c. Moreover, we will show that it is pretty much all that can be said about A(A) in ZFC, by showing that A(A) can be equal to any regular cardinal between c + and 2c and that it can be equal to 2c independently of the cofinality of 2c . This solves a problem of T. Natkaniec [10, Problem 6.1, p. 495]. We will also show that A(D) = A(A) and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.)

Density Continuity Versus Continuity, Krzysztof Ciesielski

Faculty & Staff Scholarship

Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if it is a nowhere dense F_\sigma set. The corresponding characterization for the approximately continuous functions is a first category F_\sigma set. An alternative proof of that result is given. Density …

Martin's Axiom And A Regular Topological Space With Uncountable Net Weight Whose Countable Product Is Hereditarily Separable And Hereditarily Lindelöf, Krzysztof Ciesielski

Faculty & Staff Scholarship

n the paper it is proved that if set theory ZFC is consistent then so is the following

ZFC + Martin's Axiom + negation of the Continuum Hypothesis + there exists a 0-dimensional Hausrorff topological space X such that X has net weight nw(X) equal to continuum, but nw(Y)=\omega for every subspace Y of X of cardinality less than continuum. In particular, the countable product X^\omega of X is hereditarily separable and hereditarily Lindelof, while X does not have countable net weight. This solves a problem of Arhangel'skii.