For a topological group $G$, we define $EG$ to be the infinite join of $G$, and $B$ to be the quotient of $EG$ by the left action of $G$. Explicitly $EG$ can be expressed, as a set, as $EG=\lbrace(g_{0},t_{0},g_{1},t_{1},\cdots)\in(G\times I)^{\infty}\rbrace/\sim$ such that at most finitely many $t_{i}$ are nonzero, $\sum t_{i}=1$, and the equivalence relation is induced from the join. Milnor proved that this $EG\rightarrow BG$ is a numerable principal $G$-bundle, and this is a universal bundle.
I am now studying the fact using some modern references (especially Paul Selick's "Introduction to Homotopy Theory" and Tammo tom Dieck's "Algebraic Topology"), and I can understand most of the proof, but one thing matters in my mind. Specifically, the proofs in both books say that this bundle $EG\rightarrow BG$ is numerable since $\lbrace U_{i}\rbrace_{i=0}^{\infty}$ forms a numerable cover for the bundle where $V_{i}=\lbrace(g_{0},t_{0},\cdots)\in EG\mid t_{i}>0\rbrace$ and $U_{i}:=V_{i}/G\subset BG$. But I think this cover is not locally finite, but just point-finite. Indeed I can modify the definition of $V_{i}$ to be, for instance, something like $\lbrace(g_{0},t_{0},\cdots)\in EG\mid t_{i}>2^{-(i+2)}\rbrace$, and then I can believe that $U_{i}$'s form a numerable cover. But I am afraid of the possibility that I have some misunderstandings about the topology of joins. Am I wrong? Please clarify this issue for me.
This is a great question and I have been struggling with exactly the same thing recently. Given what is stated in Dieck's proof of 14.4.3 (or rather, what is not), you are absolutely right to be worried (though it will all work out in the end). The functions $t_i:EG\rightarrow [0,1]$ do not form a partition of unity in the usual sense, since the sets $t_i^{-1}(0,1]$ only form a point-finite cover. Worse yet, the conditions in Proposition 14.4.5 seem to imply that a non-numerable bundle might possess a $G$-map into $EG$, which would only be possible if the universal bundle $EG\rightarrow BG$ were itself non-numerable. The difficulty lies in the fact that Dieck is implicitly using an equivalent definition of "numerable" that initially seems weaker. But if we look to the previous chapter of the book, we can find a magnificent fix! Namely, Lemma 13.1.7 essentially says the following (actually, it says something even stronger):
Lemma. Consider a set of continuous functions $\{u_j:X\rightarrow [0,1]\mid j\in J\}$ on a topological space $X$, such that $\{j\in J\mid u_j(x)>0\}$ is finite for every $x\in X$ and $\sum_{j\in J}u_j=1$. Then the open cover $\{u_j^{-1}(0,1]\mid j\in J\}$ is numerable. (Notice, this cover is point-finite, not locally finite! The lemma says that we can find new functions, which shrink the supports sufficiently to get a partition of unity in the usual, stronger sense.)
If you have a bundle $p:E\rightarrow X$ such that $p$ is trivializable over each $u_j^{-1}(0,1]$, then this proves that $p$ is a numerable bundle, since it is trivializable over a numerable cover. In particular, this fixes your issue with the numerability of the open cover of $EG$. However, in the course of proving Lemma 13.1.7 (which also uses the proofs of 13.1.4 and 13.1.5), you will get a new collection of functions, different from the original coordinate projections $t_i$ on $EG$. Indeed, these coordinate functions are not a numeration of the bundle and you cannot just refine the cover. You really do need to create new functions with smaller supports.