On hereditarily reversible spaces

In [15] Rajagopalan and Wilansky called a space reversible if each continuous bijection of the space onto itself is a homeomorphism. They called also a space hereditarily reversible if each its subspace is reversible. We characterize the hereditarily reversible spaces in several classes of topologicals spaces, in particular, in the class of Hausdorff spaces of the first countability and in some subclass of the class of locally finite T0-spaces relevant to digital topology. Besides we suggest different examples of (non-)reversible and (non-)hereditarily reversible spaces.