Question

prove that every subset of discrete topological space is always open and closed​

Answer 1

metric space (X,d)and every x=X, the function y (-) d(x,y) is continuous since the discrete metric is

d(x,y) ={0 x=y} { 1 , xnot equal y}

{ x} ={y=x| d ( x,y) <1/2)}

= d(x •)`1 [( 0,1/2)]