Continuous selections and purely topological convex structures

In 1983, Frank Deutsch and Petar Kenderov give some necessary and sufficient conditions for convex-valued multifunctions to have continuous approximations. Inspired by Deutsch and Kenderov's result, we introduce and characterize coherent multifunctions, investigating the relationship between lower semi-continuity and coherence. We then interpolate the lemmas behind the well-known Michael results on continuous selections. In doing so, we define a suitable and quite natural convex structure on every topological space, not just on metrizable ones. We produce a selection theorem stronger than Michael's selection theorems, both the convex-valued and the zero-dimensional version, in general considered as two independent cases in the literature.