We prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into \(([0,1]^r)^G\) is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than \(\frac{rN}{2}\) . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset \(F\subset G\) so that for a generic continuous map \(h:X\rightarrow [0,1]^{r}\) , the map \( h^{F}:X\rightarrow ([0,1]^{r})^{F}\) given by \(x\mapsto (f(gx))_{g\in F}\) is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.
