# Intersection of finite number of compact sets is compact?

Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true.

I said that this is true because the intersection of finite number of compact sets are closed. Which therefore means that it will be bounded because the intersection is contained by every set. I am not sure if this is correct.

Thank you for the help

Take $\mathbb{N}$ with the discrete topology and add in two more points $x_1$ and $x_2$. Declare that the only open sets containing $x_i$ to be $\{x_i\}\cup \mathbb{N}$ and $\{x_1 , x_2\}\cup \mathbb{N}$. (If you can’t see it immediately, check this gives a topology on $\{x_1 , x_2\}\cup \mathbb{N}$).
Now $\{x_i\}\cup \mathbb{N}$ is compact for $i=1,2$, since any open cover must contain $\{x_i\}\cup \mathbb{N}$ (it is the only open set containing $x_i$). However, their intersection, $\mathbb{N}$, is infinite and discrete, so definitely not compact.