Why a Sierpinski space is not obtainable from a pseudometric?

Answer 1

A pseudometric is a symmetric function: if ρ is a pseudometric on a set X, then ρ(x,y)=ρ(y,x) for all x,y∈X. This means that if τ is the topology on X generated by ρ, and x,y∈X, then

x∈clτ{y} if and only if y∈clτ{x}.

This is not the case with the Sierpiński space, however: in that we have 0∈clτ{1}, not 1∉clτ{0}. Thus, the Sierpiński topology on {0,1} cannot be generated by a pseudometric.