The statement I am familiar with regarding classification of vector bundles is :

Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective correspondence with the set $[X,G_n]$ of homotopy classes of maps from $X$ to $G_n$.

I am more or less comfortable with the proof of this result.

I could not **guess** how some one who has done this for the **first time** thought about it. Were there some spaces $X$ where it is immediately visible that the vector bundles over $X$ have some relation with maps from $X$ to $G_n$?

Question : How did some one guess about the possibility of vector bundles over $X$ being related to homotopy class of maps $X\rightarrow G_n$?

Pointing out a paper where this result is published definitely be useful *if* it contains some motivation how did the author(s) thought about this.

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