Michael's selection theorem states that a lower hemicontinuous multivalued map with nonempty convex closed values $\displaystyle F\colon X\rightrightarrows E$ from a paracompact space $X$ to a Banach space $E$ admits a continuous selection.
If we replace the assumption "lower hemicontinuous multivalued map with nonempty convex closed values" by one of the following assumptions, does the theorem still hold?
- "hemicontinuous multivalued map with nonempty closed values".
- "lower hemicontinuous multivalued map with nonempty connected closed values".
This is a late response I suppose, and you may have already gotten your answer elsewhere. However, the second question has some answers from (I think the second paper of) Michael’s series of papers on continuous selection. It requires that the family of $n$-connected values should also satisfy the his notion of equi-locally $n$-connectedness. He provides an example, using I think the topologist’s sine curve, where there’s no continuous selection even though the values of the multivalued mapping is path-connected.