The prediction Odysseus gives to Penelope and that Odyessus (which is himself but disguised as a beggar) will return
Odysseus' wife was Penelope, daughter of Icarius and Periboea.
he is the housekeeper for penelope and penelope is odysseus wife and odysseus is the king of ithleca
No, Penelope does not recognize that the beggar is Odysseus.
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to see if he is faithful to her as she is to him
Odysseus instructs Penelope to wait for him and to guard their home in his absence. He also warns her not to remarry and assures her that he will return to her soon.
Odysseus left Penelope bound for Troy.
Odysseus was not strictly faithful to Penelope, but was faithful to her 'in his heart.'
Odysseus told Penelope to make sure the suitors did not gain control of their household, to stay faithful to him, and to keep their son Telemachus safe.
Odysseus tells her so directly. Penelope then tests Odysseus, and he passes her test, confirming his identity.
No one did- Penelope ran a series of tests to identify Odysseus.
Odysseus who was dressed up as an old beggar told Penelope that Odysseus was alive and would return soon