Answer
Wa `alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Sister, thank you for your good question and interest to know more about Islam and its teachings.
In fact, a divorced woman cannot live with her ex-husband in the same house in case of a final irrevocable divorce. If the woman is poor or has health problems, her maharim (unmarriageable) relatives should take care of her.
Also, the State and its concerned bodies should look after poor or ill women in such cases.
Answering your question, Dr. Wael Shehab, PhD in Islamic Studies from Al-Azhar University and currently the Imam of the Downtown Toronto Masjid in Canada, states:
In case of a final, irrevocable divorce, the husband and wife turn to be strangers to each other. So, if the woman is in need of financial support or assistance owing to health problems, some of her maharim relatives—such as father, sons, brothers, etc.–should take care of this.
If there is no relative to take care of her, the community should help her. The State, moreover, should have due measures to support women in such hard situations.
It is noteworthy that an ex-husband should financially take care of his divorced wife in some specific cases such as if she is pregnant.
In all cases, a divorcee, after a final irrevocable divorce, cannot share or live with her ex-husband in in the same house or apartment.
Allah Almighty knows best.