A Muslim woman living with a married non-Muslim man bears his illegitimate child.

Answer

If the woman accepts that she is living in adultery with the non-Muslim man, then while she remains a Muslim, her Imaan dangles by a thread. It is exposed to imminent effacement. While the illegitimate child will be regarded as Muslim, his/her Imaan will be valid only on puberty. The child will become a Muslim only at the age of puberty. At that age the child should be officially introduced to Islam. Meanwhile, despite the mother being like a prostitute, the child should be regarded as Muslim and named with a Muslim name.