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Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric.

I was wondering if it will be true that: $x \rightarrow \pi^{-1}(x)$ is continuous?

Otherwise under what conditions would be true

any comment is welcome, thanks

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A few questions for clarification: First, what is your definition of a "vector bundle" over a general metric space? Second, the "mapping" $x \to \pi^{-1}(x)$ sends a point of $M$ to a vector space; what do you mean by "continuity" in this setting? Third, do you expect the presence of a Riemannian metric in $F$ to affect the answer to your question? Thanks. –  user86418 Mar 30 at 1:03

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