A facial boil that burst last week after taking antibiotics. the boil left a small hole and i am wondering if the core is gone o should i continue to clean with hydrogen peroxide solution and neosporin or what?

I : I often have patients take care of wounds like this with diluted peroxide (mixed 50/50 with water) followed by plain vaseline until the wound has healed. Without seeing your face it is hard to say whether you should continue with your regimen or not. I would follow up with whomever prescribed the antibiotic or your regular provider to get it checked out.

Will heal: It should heal without any further treatment. The small hole will eventually shrink and smooth out over time. It is possible a small scar will remain but this too should decrease w time.