Is it permissible not to greet a woman who does not adhere to Allah's law and openly commits فساد (wickedness), and is it permissible to backbite about her to warn people of her misdeeds?

Allah has forbidden backbiting and depicted its perpetrator in a repulsive image. The Prophet, peace and blessings be upon him, defined it as "your mentioning your brother with what he dislikes." However, backbiting is permissible in certain situations for a benefit, such as when one openly commits a sin, or for seeking help in enjoining good and forbidding evil, or to warn Muslims against evil, provided it is limited to the extent of the necessity.

A woman's showering at the university due to a physical necessity, if it is safe from the exposure of her awrah and from temptation, then it appears to be not forbidden. As for tabarruj (improper display of adornment), it is undoubtedly forbidden. The ruling on not greeting a woman who engages in tabarruj revolves around the benefit: if there is hope that it will lead to her reform, then greeting her is preferable; if leaving the greeting helps in her reform, then it should be left. As for her being openly sinful, this is true if she goes out displaying tabarruj among non-mahram men. It is permissible to backbite her to warn people about her if the need arises.