A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^{n+1}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.

This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,...,${\theta }_m$ of complex numbers. Specifically, let K be a number field and let f₁(z),...,$f_m\left(z\right)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) $f_j\left(z^b\right)={\sum }_{i=1}^m{f}_i\left(z\right)a_{ij}\left(z\right)+b_j\left(z\right)$ (j = i,...,m)for b ≥ 2, $a_{ij}\left(z\right)$, $b_j\left(z\right)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_j(z$) converge at z = α and the $a_{ij}\left(z\right)$, $b_j\left(z\right)$ are analytic at $z=\alpha ,{\alpha }^b,{\alpha }^{b²},...$ Then the ${\theta }_i=f_i\left(\alpha \right)$ are algebraically independent...

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ${\sum }_{n=1}^{\infty }x\left(n\right)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ${\sum }_{n=1}^{\infty }b\left(n\right)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

It is known that there is no natural Banach norm on the space $\mathrm{\mathscr{H}𝒦}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathrm{\mathscr{H}𝒦}$ space is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. On each $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space, an $F$-norm ${\parallel ·\parallel }^X$ is defined. A ${\parallel ·\parallel }^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a ${\parallel ·\parallel }^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is well-known that every...

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

We show that ${\omega }^{*}\setminus \left\{p\right\}$ is not normal, if $p$ is a limit point of some countable subset of ${\omega }^{*}$, consisting of points of character ${\omega }_1$. Moreover, such a point $p$ is a Kunen point and a super Kunen point.

For any $\alpha ,\beta >0$ with $\alpha +\beta <1$ we provide a simple construction of an $\alpha$-Hölde function $f:[0,1]\to ℝ$ and a $\beta$-Hölder function $g:[0,1]\to ℝ$ such that the integral ${\int }_0^{1}f\mathrm{d}g$ fails to exist even in the Kurzweil-Stieltjes sense.

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f\times g:X\to Y\times I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{\delta }$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and $D_j$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod }_{j=1}^k{D}_j\times I^{p+1-i}\right)$ is (i+1)-to-1 is a dense $G_{\delta }$-subset of $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ for each 0 ≤ i ≤ p.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................