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?