Let M be a compact metric space and g: M --> M be an homeomorphism C⁰-close to an expansive map of M. In general, it is not true that g is also expansive, but it still has some properties resembling the expansivity. In fact, if we identify pairs of points whose g-orbits stay nearby, both for the future and the past, we obtain an equivalence relation~. The quotient space M / ~ is a compact, metric space and g induces an expansive homeomorphism on that quotient. If M is a surface, we show that for any the connected component of the local stable (unstable) set containing is nontrivial and arc-wise connected. AMS classification scheme numbers: 58F30, 58F15, 54H20, 54F15