Prescribing the Preschwarzian in several complex variables

We solve the several complex variables preSchwarzian operator equation $[Df(z)]^{-1}D^2f(z)=A(z)$, z\in \C^n, where $A(z)$ is a bilinear operator and $f$ is a \C^n valued locally biholomorphic function on a domain in \C^n. Then one can define a several variables $f\to f_\alpha$ transform via the operator equation $[Df_\alpha(z)]^{-1}D^2f_\alpha(z)=\alpha[Df(z)]^{-1}D^2f(z)$, and thereby, study properties of $f_\alpha$. This is a natural generalization of the one variable operator $f_\alpha(z)$ in \cite{DSS66} and the study of its univalence properties, e.g., the work of Royster \cite{Ro65} and many others. M\"{o}bius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{O} play a central role in this work

A Whitney map onto the Long Arc

In a recent paper, Garc\'{\i}a-Velazquez has extended the notion of Whitney map to include maps with non-metrizable codomain and left open the question of whether there is a continuum that admits such a Whitney map. In this paper, we consider two examples of hereditarily indecomposable, chainable continua of weight $\omega_1$ constructed by Michel Smith; we show that one of them admits a Whitney function onto the long arc and the other admits no Whitney maps at all

Far points and discretely generated spaces

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points