Injective continuous images of Hamel bases
by
Andrzej Nowik
Department of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Theorem:   Under the assumption L=V we construct a Hamel bases H₁ and H₂ of reals and a continuous bijection f: H₁ → R\H₂.

Theorem:   If A is a subset of a Polish space, then the following conditions are equivalent:

A is analytic locally uncountable.

There exists a continuous surjection r: w^w → A such that ∀_{y ∈ A} r^-1[{y}] is nowhere dense.

There exists a continuous surjection r: w^w→ A such that ∀_{y ∈ A} r^-1[{y}] is nowhere dense set of size 2^w.

Date received: July 1, 2008