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Its not possible that you get a sudden eruption of white heads on face.It seems to be miliaria(prickly heat
) which develops because of sweat accumulation and heat.It can also occur on body as you have mentioned esp as red small dots.It can be mild itchy also.
You can use calamine
lotion and take a cetrizine
tablets for few days .Keep yourself in a cool environment.
If it does not go off in two days then kindly consult a dermatologist