What is the ruling on living with a non-Muslim elderly woman over sixty, due to the unavailability of university housing or the high cost of apartments, knowing that she does not bring anything Haram into the house?

It is narrated in Sahihayn and other collections that the Prophet, peace and blessings be upon him, said: "No man should be alone with a woman unless she is accompanied by a Mahram." It is not permissible for a man to live with a woman who is a stranger to him in the same apartment, even if each of them has a separate room, because the facilities are shared between them, such as the kitchen, bathroom, entrance, and so on.

The scholars have stipulated for the permissibility of cohabitation between non-mahrams that the facilities should not be shared between them. Ibn Hajar Al-Haytami said: "If a woman lives with a non-mahram in two rooms, or on different floors, or in a house with one room, it is stipulated that they do not share a facility such as a kitchen, toilet, well, rooftop, or elevator. If they share any of what is mentioned, cohabitation becomes forbidden, because it then becomes a place where unlawful seclusion (khulwa) is likely." The prohibition of seclusion is not specific to young women; it also includes elderly women.

Therefore, it is not permissible for you to live with that woman, due to the unlawful seclusion involved therein according to Islamic law.