We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations,...

We prove that if ℱ is a non-meager P-filter, then both ℱ and ${}^{\omega }ℱ$ are countable dense homogeneous spaces.

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $\varphi :X\to C_p\left(Y\right)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y\times Y$ we will use topological games ${𝒢}_1\left(H\right)$ and ${𝒢}_2\left(H\right)$ on $Y\times Y$ between two players to prove that if the first player has a winning strategy in these games, then $\varphi$ is norm continuous on a dense $G_{\delta }$ subset of $X$. It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_p\left(Y\right)$ is norm continuous on a dense $G_{\delta }$ subset of $X$.

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{\delta }$ subset of A. We will show that this class is stable under c₀-sums and ${\ell }^p$-sums of Banach spaces for 1 ≤ p < ∞.

Let $D\subset T\times X$, where $T$ is a measurable space, and $X$ a topological space. We study inclusions between three classes of extended real-valued functions on $D$ which are upper semicontinuous in $x$ and satisfy some measurability conditions.

Let $(X,d)$, $(Y,\rho )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\rho (f\left(x\right),f\left(y\right))/d(x,y);x,y\in X,x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). We investigate the minimum dimension $N$ such that every $n$-point metric space can be embedded into the space ${\ell }_{\infty }^N$ with a prescribed distortion $D$. We obtain that this is possible for $N\ge C{(logn)}^2{n}^{3/D}$, where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into ${\ell }_p^N$ are obtained by a similar method.

In this note we are going to study dense covers in the category of locales. We shall show that any product of finitely regular locales with some dense covering property has this property as well.