Cardinal-indexed classifying spaces for families of subgroups of any topological group

For $G$ a topological group, existence theorems by Milnor (1956), Gelfand-Fuks (1968), and Segal (1975) of classifying spaces for principal $G$-bundles are generalized to $G$-spaces with torsion. Namely, any $G$-space approximately covered tubes (a generalization local trivialization) is the pullback universal space indexed orbit types cardinality cover. Lie via metric model we generalize corresponding uniqueness theorem Palais (1960) Bredon (1972) compact $G$. $G$-homeomorphism proper over correspond stratified-homotopy classes maps. The former result enabled Segal's clever but esoteric use non-Hausdorff spaces. latter our own development equivariant ANR theory noncompact Applications include part classification unstructured fiber bundles locally Hausdorff connected base or fiber, as well which in certain cases other models due Lashof-May (1986) L\"uck-Uribe (2014). From categorical perspective, general $E_\mathcal{F}^\kappa G$ final object inspired formulation Baum-Connes conjecture (1994).