Sensitivity, local stable/unstable sets and shadowing

In this paper, we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary, we generalize results in [Artigue et al. Beyond topological hyperbolicity: the Lshadowing property, J. Differ. Equ. 268(6) (2020), pp. 3057–3080.] and [Carvalho and Cordeiro, Positively N-expansive homeomorphisms and the L-shadowing property, J. Dyn. Differ. Equ. 31(2) (2019), pp. 1005–1016.] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countable spaces.