# Prove X has the discrete topology, given every point is open?

I was wondering what would be sufficient to show that $X$ has a discrete topology. I know the following: $X$ is a topological space where each point $x$ is open ($\{x\}$ is open for each $x\in X$), and I want to show that $X$ has the discrete topology.

Is the following proof sufficient?

We know that each point is open. Also, any subset $U\subset X$ can be written as $\cup_{x\in U} \{x\}$, and since the union of any collection of open sets is open (by properties of a topology), it follows that any subset $U\subset X$ is open. Hence $X$ has the discrete topology.

Any help would be appreciated.

• Yep, that's just right. – Kevin Carlson Feb 12 '14 at 21:12