I'm currently studying from Husemaller Vector Bundles and I'm having some problems understanding the definition given and the conventions used by the author. I think that the book gives a definition which is very general and then the reader should specify it to the needed context. I don't understand when I have to disambiguate the latter.
For example proposition $1.5$ p.$25$ states:
Proposition 1.5: Let $\xi = (E,p,B)$ be a $k-$dimensional vector bundle. Then $p$ is an open map. The fibre preserving functions $a : E \oplus E \longmapsto E$ and $s: F \times E \longmapsto E$ defined by the algebraic operations $a(x,x') = x+x', s(k,x) = kx, k \in F$, are continuos.
What I don't understand are the following: why should I ask continuity of cross sections in the first place? the definition of vector bundle given is the following :
Definition: A $k$-dimensonal vector bundle $\xi$ over $F$ is a bundle $\xi = (E,p,B)$ together with the structure of a $k-$dimensional vector space over $F$ on each fibre $p^{-1}(b)$ such that the following local triviality condition is satisfied. Each point of $B$ has an open neighborhood $U$ and a $U-$isomorphism $h:U \times F^k \longmapsto p^{-1}(U)$ such that the restriction $b \times F^k \longmapsto p^{-1}(b)$ is a vector space isomorphism for each $b \in U$.
Here the word $U-$isomorphism seems undefined. Does the author mean homeomoprhism? the word bundle came with an other definition which doesn't seem to involve continuity, in other words a bundle is described as a triple $(E,p,B)$ where $p: E \longmapsto B$ is a map.
I don't understand what's peculiar with this definition, it seems that random maps are bundles. Could someone with more experience with me help me clarify?
In the end I don't understand what kind of structure $E$ has in order to well define sum and "scalar" multiplication, i.e $a,s$.
Any help would be appreciated.
first of all if you have maps $f:X\to S$,$g:Y\to S$,by a $S$ morphism between $h:X\to Y$ we mean that $f=h\circ g$(so the two way you can go from $X$ to $S$ should be the Same). In your example you have maps $E_{|U}\to U$,$U\times V\to U$ so you can talk about a $U$ morphism between $E_{|U}$ and $U\times V$(the point of a $S$ morphism $h$ is basically that you want $h$ to sends fiber over $s$ in $X$ to the fiber over $s$ in $Y$ ).
When you define a vector space, you don't mention continuity because you don't even have a topology, but it is an easy exercise that if you have a topological field(like $\mathbb{R},\mathbb{C}$) if you put the product topology on $V$, the sum and scalar product become continuous.
about the relationship between definition and proposition: when you have $p: E\to B$ by definition you have a sum and scalar product on each fiber so you get maps $E\times E\to E$, and $F\times E\to E$ and these are continuous because $E$ locally looks like $U\times V$ and for $U\times V$ you know that these maps are continuous(by previous paragraph).
the idea of the bundle is that it is a space $E$ over $B$(meaning that there is a map $E\to B$) that locally looks like the product of $B$ with a fixed space. if this fixed space is a vector space, we call $E$ a vector bundle. but you can also talk about a circle-bundle or a $F$-bundle for any fixed space you pick.