If $\{K_n\}_n$ is a family of compacts such that $\bigcap_nK_n=\varnothing$, then $\exists N\in\mathbb{N}$ such that $\bigcap_{i\le N}K_i=\varnothing$

Question

I was reading Bogachev's Measure Theory book, when I stumbled upon this part:

The specification "more generally, in a topological space" made me wonder if the same result can be proved for a family of compact sets in a generic topological space $(X,\tau)$. In this proof Bogachev uses some specific properties of Hausdorff/metric spaces (like for example, the fact that the $E_n$ are compact, which is not true in general for a non-Hausdorff topological space, or the sequential compactness), and I tried to see if I could generalize it. I managed to bypass the "$E_n$ are compact" part pretty simply, but to complete the proof I need the following property: if $(x_n)_n\subseteq K$ is a sequence in a compact set $K$, and $x$ is an accumulation point for that sequence, then $x\in K$.

The problem is that, intuitively, I don't think that such a property holds in general, since it seems related to the properties of a closed set (instead of compact). So my question are:

• Does this property hold in general?
• Is it possible (maybe using a different approach) to prove the result for a generic topological space? Or is it necessary to make an extra hypothesis (like the space is Hausdorff/metric)?
• If this result doesn't hold in general, is it possible to construct a counterexample in which the property fails (i.e. a compact family which is not a compact class)?

Answer 1

This is not true in general.

For example, consider $\mathbb{N}$ with the cofinite topology (whose open sets are $\varnothing$ and $U\subseteq \mathbb{N}$ such that $\mathbb{N}\setminus U$ is finite). Then every subset is compact (an exercise for you), so setting $K_n = \{m\in \mathbb{N}\mid m\geq n\}$ gives a family of compact sets with empty intersection, but with all finite intersections non-empty.

The statement is true in any space where all compact sets are closed. For example, this is true in any Hausdorff space.