# Is this metric space compact or not under usual metric and discrete metric

Let $X = (0,1]$, $d_1$ be the usual metric on $X$, and $d_2$ be a discrete metric on $X$. Which of the following is true:

1. $(X,d_1)$ is compact, but $(X,d_2)$ is not.
2. $(X,d_2)$ is compact, but $(X,d_1)$ is not.
3. Both $(X,d_1)$ and $(X,d_2)$ are compact.
4. Neither $(X,d_1)$ nor $(X,d_2)$ is compact.
• What's your definition of compactness? – Ravi Sep 10 '17 at 17:17

Hint: For the usual metric consider the open cover consisting of the sets $(1/n,1]$ for $n \geq 1$.