In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a topological group then any $G$-resolution $X$ is a principal $G$-bundle over $B_{G} = X /G$ with the action of $G$ in $X$ as a principal map.

In the article no proofs are given so I would like to know of any reference that would point me into a direction to prove this statement.

A $G$-resolution is defined as a filtered $G$-spaces $X = \cup_{i=0}^{\infty} X_{i}$ such that it satisfies:

1. Acyclicity condition: every level $X_{n}$ is contractible in $X_{n+1}$ to a point $x_{0}$ on the bottom $X_{0}$.

2. Free condition: for every $n$ exists a closed subspaces $D_n$ between the levels $X_{n-1}$ and $X_{n}$ such that the action induces a relative homeomorphism $(D_{n}, X_{n-1}) \times G \rightarrow (X_{n}, X_{n-1})$.

In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a topological group then any $G$-resolution $X$ is a principal $G$-bundle over $B_{G} = X /G$ with the action of $G$ in $X$ as a principal map.

In the article no proofs are given so I would like to know of any reference that would point me into a direction to prove this statement.

A $G$-resolution is defined as a filtered $G$-space $X = \cup_{i=0}^{\infty} X_{i}$ such that it satisfies:

1. Acyclicity condition: every level $X_{n}$ is contractible in $X_{n+1}$ to a point $x_{0}$ on the bottom $X_{0}$.

2. Free condition: for every $n$ there is a closed subspace $D_n$ between the levels $X_{n-1}$ and $X_{n}$ such that the action induces a relative homeomorphism $(D_{n}, X_{n-1}) \times G \rightarrow (X_{n}, X_{n-1})$.