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I am sorry if this question is too trivial but I couldn't find the answer.

Who did first classify topological principal $G$-bundles for some topological group $G$? So, I mean that equivalence classes of principal $G$-bundles on a nice space $X$ are 1-to-1 to homotopy classes [X,BG] where $BG$ denotes the classifying space of $G$.

My first guess was that this is due to Steenrod but then I looked up that Milnor did his construction on classifying spaces in 1956 what is much later.

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when G is the orthogonal group I think it was most likely Hassler Whitney since the techniques of the proof are very similar ro the weak Whitney embedding theorem. –  Ryan Budney Jul 18 '12 at 14:29

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The first paragraph of the following article hints that Steenrod, Whitehead, Chern and Sun indepentently arrived at the fact that -- provided that the universal bundle exists -- it classifies principal $G$ bundles. (I don't have a subscription so I can't check the references for details, in particular the date of publication...)

S.-T. Hu: The Equivalence of Fiber Bundles, Ann. Math. (2) 53 1953, http://www.jstor.org/discover/10.2307/1969542?uid=3737864&uid=2129&uid=2&uid=70&uid=4&sid=49501340154377

A partial result (the fibre is $S^n$) was proved earlier by Steenrod (I believe you can find the general result in Steenrod's book on fibre bundles, which was published in 1951). Milnor, however, seems to be the first one to establish the existence of universal bundles.

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