  In order to establish Fredholm theory on stratified topological Banach
manifolds in Gromov-Witten theory, we have introduced flat structures on such
manifolds in [L4]. Such a structure is obtained from local flat coordinate
charts. The transformations between these charts are only continuous in
general. The purpose of this paper is two-fold : firstly to show that on the
topological Banach manifolds and bundles appeared in Gromov-Witten and Floer
type theories, there are enough smooth functions and sections viewed in any
admissible charts and trivializations; secondly to demonstrate some finer
aspects about the weakly smooth sections.
