Consider a $C^{k}$ map / flow on a plane, with a hyperbolic fixed point at the origin $0$, with a $1$-dimensional stable subspace, and a $1$-dimensional unstable subspace. By the stable /unstable manifold theorem, there exist $1$-dim stable and unstable manifolds $W^{s,u}(0)$ of $0$. Further, suppose that these manifolds coincide in a figure-8 homoclinic loop $\Gamma \subset W^{s}(0) \cap W^{u}(0)$.
Many authors refer to the homoclinic loop as "homoclinic manifold".
My question is, in what sense is this a manifold - since strictly speaking, figure 8 is not a topological / differentiable manifold (due to the point of self-intersection)?
The only way this can be if the origin is removed from the domain of definition of $\Gamma$...So does this mean that the domain of definition of $W^{s,u}(0)$ also excludes $0$?
Or in the context of smooth dynamical systems, the terms stable / unstable / homoclinic manifolds also refer to manifolds with certain "singularities"?