Timeline for Existence of enough local sections

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Mar 15, 2016 at 11:58	comment	added	Simon Henry		I know only one result in this spirit: if $X$ is a locally paracompact topological space and $G$ is a semi-continuous bundle of Banach space (so one forget compactness, keep the "open" map and in addition to your axioms, each fiber is a real banach space, the action of $\mathbb{R}$ is globally continuous and the norm function is upper semi-continuous) then it has enough local sections. But the proof uses partition of unity and hence the multiplication by $\mathbb{R}$ a lot, so it is not clear that the idea of this results can be adapted here.
Mar 15, 2016 at 9:32	history	asked	user1688	CC BY-SA 3.0