Definition of vector bundles - what is meant by a structure of a real vector space?

Question

I am currently reading John Lee's book about Differential geometry and I don't fully understand his definition of vector bundles. Unfortunately, Wikipedia didn't help.

A vector bundle of rank $k$ is a continuous map $\pi\colon E\to M$ with the following property (among others):

For each $p\in M$, $\pi^{-1}(p)$ is endowed with the structure of a $k$-dimensional real vector space.

On page 13 of "Introduction to Smooth Manifolds", he defines a smooth structure to be a smooth atlas, so I thought that a structure might be an atlas. But even is if this is true, I don't see what he means by the addition "of a $k$-dimensional real vector space".

Answer 1

It's structure in the general mathematical sense of "adding something" to a set to get a richer object:

• To endow a set $G$ with a group structure is to pair it with an operation $*$ such that $(G,*)$ is a group;
• To endow a set $V$ with a vector space structure is to pair it with operations $+, \cdot$ satisfying the vector space axioms;
• To endow a set $M$ with a smooth structure is to pair it with a smooth atlas.

In this case, what is meant is that part of the definition of a vector bundle is the choice of a $k$-dimensional vector space structure on each fiber $\pi^{-1}(p),$ i.e. $+$ and $\cdot$ satisfying the vector space axioms are defined on each $\pi^{-1}(p).$