Everywhere surjections and related topics. Examples and counterexamples

This paper deals with everywhere surjections, i.e. functions defined on a topological space whose restrictions to any non-empty open subset are surjective. We introduce and discuss several constructions in different contexts; some constructions are easy, while others are more involved. Among other things, we prove that there is a vector space of uncountable dimension whose non-zero elements are everywhere surjections from Q to Q; we give an example of an everywhere surjection whose domain is the set of countably infinite real sequences; we construct an everywhere surjective linear map from the Cantor set into itself. Finally, we prove the existence of functions from R to R which are everywhere surjections in stronger senses.