Is the Milnor construction contractible Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$.

Is $E_G$ contractible?

I mean it is clear that $E_G$ is weakly contractible, but I dont see why it should be contractible if $G$ is not a CW-space.
 A: Oliver Straser is correct in that Milnor himself in 1956 [1] only showed his model for $EG$ is weakly contractible (that is, all its homotopy groups vanish) with his coarse topology on the join.  (For any compact Hausdorff spaces $X$ and $Y$, by the tube lemma, note that Milnor's $X \circ Y$ equals the quotient-topology join $X \ast Y$.)
Historically, for any topological group $G$, the first proof in the literature of the contractibility of Milnor's $EG := G^{\circ \aleph_0}$ is contained within the two-page proof of Theorem 8.1 in Dold's 1963 article [2].
A few years later in 1966, tomDieck in Section 7 of [3] offers a more conceptual argument.
Contemporarily, I agree that the comment of Oldřich Spáčil is an acceptable answer, based on the simplified proof that tomDieck later gives in Proposition 14.4.6 of his 2008 book [4], which rehashes his earlier works.
[1] : John Milnor, Construction of universal bundles II, Annals Math 63(3):430–436, 1956.
[2] : Albrecht Dold, Partitions of unity in the theory of fibrations, Annals Math 78(2):223–225, 1963.
[3] : Tammo tomDieck, Klassifikation numerierbarer Bündel, Arch Math (Basel) 17:395–399, 1966.
[4] : Tammo tomDieck, Algebraic Topology, 2008.
