  We prove that local stable/unstable sets of homeomorphisms of an infinite
compact metric space satisfying the gluing-orbit property always contain
compact and perfect subsets of the space. As a consequence, we prove that if a
positively countably expansive homeomorphism satisfies the gluing-orbit
property, then the space is a single periodic orbit. We also prove that there
are homeomorphisms with gluing-orbit such that its induced homeomorphism on the
hyperspace of compact subsets does not have gluing-orbit, contrasting with the
case of the shadowing and specification properties, proving that if the induced
map has gluing-orbit, then the base map has gluing-orbit and is topologically
mixing.
