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A topological space is compact if and only if for every collection of closed subsets of it such that any finite intersection of these sets is not empty the intersection of all of these subsets is not empty.

I heard that this can be intuitively thought of as closed sets being able to locate points. Does this have any actual meaning? And if so, could someone explain it?

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  • $\begingroup$ It looks like this is formally equivalent to the usual open-cover definition by (1) instead of quantifying over all families of open set, speak of families of their complements (which are closed), plus (2) take the contrapositive of the implication under that quantifier. $\endgroup$ – Henning Makholm Aug 22 '18 at 19:05
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I believe that the idea behind "closed sets can locate points with compact spaces" is the following. If $X$ is any topological space and a point $x\in X$ then there is the collection $\mathcal{C}_{x}$ of closed subsets of $X$ that contain $x$. Naturally this collection $\mathcal{C}_{x}$ has the property you mentioned (which I've seen referred to as the 'finite intersection property' or 'FIP'). Of course $x\in\bigcap\mathcal{C}_{x}$. Thus we might say that $x$ is 'located' or perhaps 'specified' by the collection of closed sets that contain it.

The question is whether or not we can go the other way. If we have a collection of closed sets that satisfies the finite intersection property does it necessarily specify a certain point in $X$? In general of course the answer is no. If $x=(0,1)$ is given its standard topology then defining $A_{n}=(0,\frac{1}{2n}]$ for each $n\in\mathbb{N}$ gives a collection of closed sets in $X$ with the finite intersection property whose intersection is empty. This might feel like an odd phenomenon in general. It seems like the set $\{A_{n}\}_{n\in\mathbb{N}}$ is very much like a cofinal subset of some $\mathcal{C}_{x}$. This idea appears in compactification theory. For example the Wallman compactification of a Hausdorff space $X$ is construction by taking the collection $\gamma X$ of all closed ultrafilters on $X$ and defining a topology on $\gamma X$ in the following way:

For each closed $D\subseteq X$ define

$$D^{*}:=\{\mathcal{F}\in\gamma X\mid D\in\mathcal{F}\}$$

Then defining $\mathscr{C}$ to be the collection of all such $D^{*}$ (where $D$ is closed) we have that $\mathscr{C}$ is a basis for the closed sets of a topology on $\gamma X$. Then the mapping $h:X\rightarrow\gamma X$ which maps each point $x\in X$ to the unique ultrafilter converging to it is an embedding. Moreover, $\gamma X$ is a compact (Hausdorff) space.

I would summarize in the following way: in general topology you don't really need to think about points that often. You only need to think about neighbourhood systems. The closed sets containing a point determine that point and other points topological indistinguishable from it. In a compact space, every "nice" collection of closed sets determines at least one point.

*This characterization of the Wallman compactification is from Stephen Willard's "General Topology".

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