I understand the first two points of this definition. However I need assistance in understanding further the sentance: It is required that the topology of $E_x$ as a subspace of $E$ coincides with its standard topology as a real vector space.
In abstraction the definition makes sense, but I would like to know what would the topology on $E_x$ look like as a vector space?
Since the pullback bundle is a bundle, is it possible that if we have a bundle $\pi_E: E \to X$ over a topological space $X$ and a map $f : X \to Y$ we can construct a bundle say $\pi_F:F \to Y$ over a topological space $Y$ ? Essentially my question is that is it possible to construct such a bundle using images ? what properties should the homomorphism $h: E \to F$ admit other than continuity?
Is there a sense of talking about a bundle being contained in another bundle?
For the third question, we can talk about topologies being contained in another, I want to understand if it is also possible for vector bundles.
The link for the paper: https://ncatlab.org/nlab/files/wirthmueller-vector-bundles-and-k-theory.pdf