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If Paul allow Timothy to go on into Timothy's mission being " unsaved " then Paul would have been much unwise and stupider than I thought.
Paul is giving Timothy a exhortation, a encouragement to keep in the word and study it and not be a slackhard.
That gift ? it could be one of the gifts of the Holy Spirit or it could have been a special gift that Timothy had in him that Paul recognized and he told Timothy to keep it going and not let it become stagnant.
It could have been the gift of knowlegde and having a great understanding of the scripture for Paul said to give him self to it.
Take heed of himself and the doctrines ?
He's telling Timothy to not negelct what he has been taught, when he has seen and learned, and don't allow anyone to lead him assray in that doctine therefore he will be saved from false teaching and those around him.
The word save here ? to save himself from those who would try to corrupt that doctrine in what he has been brought up in and corrupt the message of the Gosple.
It's not the same meaning as the word we use as " saved, or born again " but rather save as in being untainted from those who would come in and corrupt the Gosple, the doctrines and teachings of Paul that Paul entrusted to Timothy.

Yeah, yeah, I know, my engish and spelling is terrible.

Correction " He's telling Timothy to not neglect what he has been taught, WHAT he has seen and learned, and don't allow anyone to lead him astray in that doctrine therefore he will be saved from false teaching and those around him.