We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes...

Let us denote by $\Phi (\lambda ,\mu )$ the statement that $𝔹\left(\lambda \right)=D{\left(\lambda \right)}^{\omega }$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $ℂ$ in $𝔹\left(\lambda \right)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V}\subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and...

A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f:ℝ\to ℝ$ is of the first Baire class if and only if for each $ϵ>0$ there is a sequence of closed sets ${\left\{C_n\right\}}_{n=1}^{\infty }$ such that $D_f={\bigcup }_{n=1}^{\infty }{C}_n$ and ${\omega }_f\left(C_n\right)<ϵ$ for each $n$ where $${\omega }_f\left(C_n\right)=sup\left\{\right|f\left(x\right)-f\left(y\right)|:x,y\in C_n\}$$ and $D_f$ denotes the set of points of discontinuity of $f$. The proof of the main theorem is based on a recent $ϵ$-$\delta$ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

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 show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

For a finite group $G$ denote by $N\left(G\right)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N\left(G\right)=N\left(L\right)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z\left(G\right)=1$ and $N\left(G\right)=N\left(A_i\right)$ is necessarily isomorphic to $A_i$, where $i\in \{2p,2p+1\}$.

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

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 $𝔠$.

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

We prove that each linearly continuous function $f$ on ${ℝ}^n$ (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for $f$ on an arbitrary Banach space $X$, if $f$ has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such $f$ on a separable $X$ is continuous at all points outside a first category set which is also null in any usual...