A function is defined to be hardly open provided that the inverse image of each dense subset of the codomain that is contained in a proper open set is dense in the domain. This form of weak openness is shown to be strictly between near feeble openness and somewhat openness. Characterizations and properties of hardly open functions are presented.
BAKER, C. W. (1996) "WEAK FORMS OF OPENNESS BASED UPON DENSENESS," Turkish Journal of Mathematics: Vol. 20: No. 3, Article 9. Available at: https://journals.tubitak.gov.tr/math/vol20/iss3/9