The Fibonacci cube $Γ_n$ is is the graph whose vertices are independent subsets of the path graph of length $n$, where two such vertices are considered adjacent if they differ by the addition or removal of a single element. Klavžar [1] suggested considering the infinite Fibonacci cube $Γ_\infty$ whose vertices are independent subsets of the one-way infinite path graph with the same adjacency condition. We show that every connected component of $Γ_\infty$ is asymmetric (has no nontrivial automorphism) and no two connected components of $Γ_\infty$ are isomorphic. This follows from our results on a further generalization $Γ_G$ where $G$ is a simple, locally finite hypergraph with no isolated vertices.