When is the critical manifold of a dynamical system an attractor?

Question

In the study of autonomous dynamical systems of the form $\dot{x}=f(x)$ with $f : \mathbb{R}^n \to \mathbb{R}^n$, one has the notion of a hyperbolic equilibrium point $x^e : f(x^e)=0$, $Re(\lambda_i) \neq 0$ $\forall i$ where $\lambda_i$ are the eigenvalues of $Df(x^e)$ (the Jacobian of $f$ at $x^e$). The equilibrium is then characterized as exponentially stable if $Re(\lambda_i) < 0$ $\forall i$ or unstable otherwise (http://www.scholarpedia.org/article/Stability#Hyperbolic_equilibria). Sometimes I have seen this result also combined with the assumption that the equilibrium point is isolated, and other times not. What is the difference and what is the correct characterization of stability for a critical manifold of equilibrium points? Is there some refinement of a hyperbolic set where the set is point-wise invariant rather than only set-wise invariant?