What is the difference between indiscrete metric space and indiscrete topological space? What is the difference between indiscrete metric space and indiscrete topological space? I'm confusing both are same. I have also read one theorem that every metric space is Hausdorff but here both are not Hausdorff. This create lot of confusion to me

If an indiscrete topological space $X$ is metrizable then $X$ is necessarily a singleton whereas it can be any set without metrizability. Proof: Fix $x$. $B(x,r)=X$ for any $r>0$ so $d(y,x)