HIV (human immunodeficiency virus) is a virus that cannot survive out of the body for longer than several minutes. It also needs humidity to survive. Drying the specimen in which it is found (blood, semen, saliva, and tears) reduces significantly the number of viral particles.
Given the data you provided (your child drinking from the same bottle as your neighbor who has AIDS, the bottle previously put in refrigerator, your child losing a tooth a day before, not sure if your neighbor has sores or bleeding in mouth), the chances for transmission are reduced almost to zero if the bottle stayed in the refrigerator for more than several minutes.
The sores or the bleeding in the mouth are not always necessary for transmission, because the virus can be found in the saliva, although whether the amounts are enough for transmission is still a matter of discussion among scientists.
You might want to test your child just to calm yourself. A test is possible because one month has passed. Tests at three months after possible exposure are also available.
You might want to consult with a specialist for infective diseases for more information about this problem.
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