Let X be a metric space with metric d prove that any onepoin

Let X be a metric space with metric d. prove that any one-point set is closed

Solution

one point set {x}is certainly closed,

because their complements are open:

given any aX{x} for aX{x}, let =|ax|

Then x(a,a+)

so (a,a+)X{x}

hence X{x} is open,

so {x} is closed.

any one point set is closed.