[Math] Why discrete topology is power set of a set

Question

Let $X$ be a set, then the discrete topology $T$ induced from discrete metric is $P(X)$, which is the power set of $X$

I know $T \subset P(X)$, but how do we know $T=P(X)$

Thank you!

Answer 1

From the definition of the discrete metric, taking a ball of radius $1/2$ around any element $x \in X$ gives you that $\{x\} \in T$.

Let $A \subset X$ be an element of $P(X)$. Then $$A = \bigcup_{a \in A}\{a\} \in T$$ since any union of elements in $T$ is an element of $T$. This proves that $P(X) \subseteq T$, and you already have $T \subseteq P(X)$, hence $T = P(X)$.