An example of a $T_1$ topological space, for which not all compacts are Borel sets.

Question

I am looking for a $T_1$ topological space for which not all compacts lie in the induced Borel $\sigma$-algebra.

I already have an example for a $T_0$ space with this property, namely the prime spectrum of the ring $\mathbb{Z}$ with the Zariski topology, but it is sadly not $T_1$. There is a similar question here, where in one of the replies someone suggests looking at the affine space $\mathbb{C}$ again with the Zariski topology, but I am not sure I get why that would work.

Any insight would help out quite a lot!

Answer 1

For any set $X$ the cofinite sets defines a $T_1$ topology on $X$. The borel sets with respect to the cofinite topology are the countable and cocountable sets. Take $X$ to be an uncountable set and $Y\subseteq X$ such that $Y$ and $X\setminus Y$ are both uncountable. We have that $Y$ is not Borel and it is compact since any subset of the cofinite topology will be compact.