Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions ${\displaystyle \{W^S\left(x\right)\mid x\in X\}}$ and ${\displaystyle \{W^u\left(x\right)\mid x\in X\}}$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, ${\displaystyle W^{\sigma }\left(x\right)}$ contains a nondegenerate subcontinuum ${\displaystyle A_x}$ containing x with ${\displaystyle diam{A}_x\ge \varrho }$, and if x,y ∈ C and x ≠ y, then ${\displaystyle W^{\sigma }\left(x\right)\ne W^{\sigma }\left(y\right)}$. For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), ${\displaystyle W^u(˜x)}$ is equal to the arc component of (G,f) containing ˜x, and ${\displaystyle dim{W}^s\left(W^x\right)=0}$.