A new class of functions called “$R_z$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_z$-supercontinuous functions properly includes the class of $R_{\mathrm{cl}}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\mathrm{cl}$-supercontinuous ($\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983), and is...

As a special case of the general question - “What information can be obtained about the dimension of a subset of ${ℝ}^n$ by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in ${ℝ}^3$ each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in ${ℝ}^n$ whose images contain open sets, expanding on a result of Borsuk.

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on ${\left[\omega \right]}^{\omega }$, ${\left(s\right)}_0$ is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is...

Let $(X,\rho )$, $(Y,\sigma )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\sigma (f\left(x\right),f\left(y\right))/\rho (x,y)$; $x,y\in X$, $x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\epsilon >0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\to Z$ ($Z$ arbitrary metric space) with $dist\left(f\right)<K$ one can find a mapping $g:X\to Y$, such that both the mappings $g$ and ${f|}_{g\left(X\right)}$ have distortion at...

Various ordinal ranks for Baire-1 real-valued functions, which have been used in the literature, are adapted to provide ranks for Baire-1 multifunctions. A new rank is also introduced which, roughly speaking, gives an estimate of how far a Baire-1 multifunction is from being upper semicontinuous.

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T: C*(X,E) → C*(Y,F) is a biseparating...