On the isometrization of groups of homeomorphisms

Let $G$ be a group of homeomorphisms of a topological space $X$. $G$ is $\mathit{\text{(properly)}}\mathit{\text{isometrizable}}$ if there exists a $G$-invariant (proper) gauge structure on $X$. $G$ is $\mathit{\text{equiregular}}$ if for every $x\in X$ and every open neighborhood $U$ of $x$ in $X$ there is an open neighborhood $V$ of $x$ in $X$ such that $cl(V)\subset U$ and every $y\in X$ has an open neighborhood $N_y$ with the property that for every $g\in G$, if $g(N_y) \cap cl(V) \neq \emptyset$,then$g(N_y) \subset U$\mathrm{.}$G$is$$\mathit{\text{nearly}}\mathit{\text{proper}}$$ifforallcompactsubsets$A$and$B$of$X$,$cl$($\bigcup{g(A):g\in G$and$g(A)\cap B \neq \emptyset} ) is compact.G$ $\mathit{\text{acts}}\mathit{\text{properly}}\mathit{\text{on}}$ $X$ if for all compact subsets $A$ and $B$ of $X$, the subset $G_{A,B}$ = { $g\in G:g(A)\cap B\mathrm{\ne }\mathrm{\varnothing }$ } is compact when $G$ is endowed with the compact-open topology. THE ISOMETRIZATION THEOREM: If $X$ is a Hausdorff space and $G$ \ $X$ is a paracompact regular space, then: $G$ is isometrizable if and only if $G$ is equiregular. THE PROPER ISOMETRIZATION THEOREM: If $X$ is a locally compact $\sigma$-compact Hausdorff space and $G$ \ $X$ is a regular space, then: $G$ is properly isometrizable if and only if $G$ is equiregular and nearly proper. The PROPER ISOMETRIZATION THEOREM has the following corollary. THEOREM OF ABEL-MANOUSSOS-NOSKOV: If $X$ is a locally compact $\sigma$-compact Hausdorff space and $G$ acts properly on $X$, then $X$ is properly isometrizable.