From what action does Oedipus’s fame come?

Oedipus of Oedipus Rex is famous for two actions the oracle foretold about him: he killed his father and married his mother. He didn't mean to do either one, but the point of the play is that humans can neither fool the gods nor dodge their fates.
When Oedipus's parents learned the prophecy that he would kill his father and marry his mother, they tried to have him exposed as an infant so that he would die. He was, however, saved, and he grew up with people in Corinth that he thought were his parents. When he discovered the prophecy, he left Corinth, horrified at the thought he could harm his parents.
On the road to Thebes, he killed a man, not knowing it was his father, the King of Thebes. When he arrived in Thebes, he married the king's widow, his mother. It was not until later that he came to realize what he had done.