Countable approximation of topological $G$-manifolds, II: linear Lie groups $G$

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving’s dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon–Floyd theorem from compact groups $G$ (1960).