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Once the relative humidity of a mass of air becomes 100%, then if the temperature falls it cannot hold all of the water vapour within it. The excess water vapour will then condense into cloud or fog or, if in contact with objects on or near the ground, will form dew or hoar frost.
When observing successive weather reports (METARs), a reducing gap between the actual temperature and the Dew Point temperature gives an indication of impending low visibility conditions and the posibility of fog.