# Numerable covers from the point of view of Grothendieck topologies

Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible total space $EG$. It classifies principal $G$-bundles in the sense that for any paracompact space $B$, isomorphism classes of locally trivial principal $G$-bundles are in natural bijection with homotopy classes of maps from $B$ to $BG$.

More generally, if $B$ is any topological space, homotopy classes of maps from $B$ to $BG$ are in bijection with isomorphism classes of principal $G$-bundles for which there exists a numerable open cover $(U_i)_i$ of $B$ such that the bundle is trivial on each $U_i$. This means that there exists a partition of unity subordinate to that cover. I believe that this is due do A. Dold (Partitions of unity in the theory of fibrations, Annals of math., 1965).

Now observe that the family numerable covers defines a Grothendieck topology on a topological space. Hence, this result of Dold falls within the circle of thoughts (development of étale cohomology on algebraic varieties, sites, toposes,...) that was actively developed by Grothendieck at the time Dold wrote his paper.

My question is whether there has been any kind of connections between these two groups of mathematicians or, in the contrary, whether this is a mere coïncidence.

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I think you mean numerable, not numerical. – David Roberts Dec 21 '12 at 9:20
You're right, let me edit the question... – ACL Dec 21 '12 at 11:11