No, Castor is not the son of Zeus.
Castor is one of the twin sons of Tyndareus, the King of Sparta, and Leda.
His twin brother is Pollux, who is the son of Zeus. This is because Zeus seduced Leda in the form of a swan.
Castor and Pollux are known as the Dioscuri, meaning "sons of Zeus." While Castor was mortal, Pollux was immortal due to his divine parentage.
The Dioscuri were renowned for their bravery and loyalty to each other. After Castor's death, Pollux begged Zeus to allow him to share his immortality with his brother. Zeus granted this request, allowing them to spend half their time in the Underworld and half in the heavens.
This unique story makes the Dioscuri a fascinating example of the complexities of Greek mythology and the power of familial bonds.