Fibrations and global injectivity of local homeomorphisms

Given X a path connected space and g:X→R a local fibration on its image, we prove that g satisfies a kind of deformation and consequently we obtain the path connectedness of its level sets. Then we provide global injectivity and invertibility theorems for local homeomorphisms f:X→Rn. These generalize known analytical results such as those given by Balreira and by Silva and Teixeira.