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According to Global (smooth) sections of a tensor product of vector bundles on a smooth manifold, there exists an isomorphism of $C^{\infty}(M)$-modules $$\Gamma(V,M)\otimes_{C^{\infty}(M)}\Gamma(W,M)\rightarrow\Gamma(V\otimes W,M)$$ for a smooth manifold $M$ and two smooth real vector bundles $V,W$ over $M$. My question is what can we say about the vector space of all continuous sections of the tensor product bundle $V\otimes W$ over $X$ $$\Gamma(V\otimes W,X)$$ when $X$ is an arbitary topological space and $V,W$ are merely real vector bundles?

Do we have a similar result as above?

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    What do you mean by $X$ arbitrary? any topological space? and what how do you take sections of a vector space? do you mean sections of the trivial bundle $X\times(V\otimes W)\to X$? Finally, what kind of results do you expect or need? – Jackozee Hakkiuz Sep 02 '22 at 14:30
  • @JackozeeHakkiuz Thank you, edited –  Sep 02 '22 at 18:21
  • What kind of sections? In the smooth manifold case, the result applies to smooth sections. In the case you are asking about, are you considering continuous sections? – Thorgott Sep 02 '22 at 18:50
  • @Thorgott Thank you, edited –  Sep 03 '22 at 12:11
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    Ok, so there always is a map $\Gamma(V,X)\otimes_{C^0(X)}\Gamma(W,X)\rightarrow\Gamma(V\otimes W,X)$ of $C^0(X)$-modules. If $X$ is compact hausdorff, a partition of unity similar to the one in the smooth case shows this map is an isomorphism. I haven't though about the more general cases. – Thorgott Sep 03 '22 at 14:43

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