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Let $X$ be a topological space, and let $\operatorname{CL}(X)$ be its hyperspace. That is, $\operatorname{CL}(X)$ is the set of closed subsets of $X$, equipped with the minimal topology so that the canonical map $$x \mapsto \overline{\{x\}}$$is a homeomorphism onto its image.

Typically, the base space is assumed to be Hausdorff (or at least $T_1$), so that the closure of a singleton is the singleton itself. However, the definition of a hyperspace is perfectly suitable when the space is not Hausdorff, and surely this comes in handy sometimes.

What is a good example of a hyperspace $\operatorname{CL}(X)$ where the base space $X$ is non-Hausdorff?

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The first example of a non-Hausdorff space that comes to mind is the Zariski topology on an algebraic variety. What does the hyperspace of such a topology look like? –  Noah S Feb 14 '13 at 6:44
The more standard format on MO for a tag wouuld be 'hyper-topology'. In case you edit the question, you might consider renaming this new tag (via simply changing it). –  quid Feb 14 '13 at 9:55
I think people look at this when X is a poset with the topology in which upper sets are open. Then the closed sets are lower sets. –  Benjamin Steinberg Feb 14 '13 at 11:38
@quid: I've never seen it hyphenated. On the other hand, I would prefer the tag 'hyperspaces'. –  François G. Dorais Feb 14 '13 at 17:30
An easy example. Let $X=\set{a, b, c, d\}$ and $T=\set{\emptyset, X, \set{a,b\}\}$, then $(X, T)$ is a no-Hausdorff space. –  Ali Feb 15 '13 at 4:33
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2 Answers

up vote 1 down vote accepted

I've made use of this in some (unpublished) work in connection with a formalism for discussing so-called geometric limits of holomorphic dynamical systems. The details of the specific application are not so relevant, but I've copied the statement here to give a sense of how what is essentially Fell's Theorem yields a useful compactness statement that was otherwise not possible to even formulate this precisely. Todd and Benjamin's comments are relevant. What saves the day is the fact that the original space, while not Hausdorff, is still sober.

A holomorphic dynamical system on a complex manifold $X$ is any collection of open analytic maps, from open subsets to $X$, containing the identity and all implied restrictions and compositions. We say that the systems ${\cal F}_\eta$ converge geometrically to the system $\cal F$ whenever

$$\liminf {\cal F}_\eta ={\cal F}= \limsup {\cal F}_\eta$$ where $\liminf$ and $\limsup$ are given by the prescription:

$$\liminf {\cal F}_\eta = \{f: f_\eta\rightarrow f \mbox{ for some } f_\eta \mbox{ chosen from } {\cal F}_\eta\}$$

$$\limsup {\cal F}_\eta = \{f: f_{\eta_\kappa}\rightarrow f \mbox{ for some } f_{\eta_\kappa} \mbox{ chosen from some } {\cal F}_{\eta_\kappa}\}.$$

By $f_\eta\rightarrow f$ we mean uniform convergence on compact subsets of converging domains: that is, the domain of $f$ contains a given compact set if and only if the domain of $f_\eta$ eventually does. A system $\cal F$ is closed if it contains every $g$ such that $f_\eta\rightarrow g$ for some $f_\eta\in{\cal F}$.`

We denote the set of holomorphic dynamical systems on $X$ by $HDS(X)$, and the subset of closed holomorphic dynamical systems by ${\bf HDS}(X)$.


(1) There is a unique topology on ${\bf HDS}(X)$ such that ${\cal F}_\eta\rightarrow{\cal F}$ if and only if $\liminf {\cal F}_\eta ={\cal F}= \limsup {\cal F}_\eta$.

(2) The space ${\bf HDS}(X)$ is compact and Hausdorff.

(3) If $X$ has countably many components then ${\bf HDS}(X)$ is second countable and metrizable.

The space ${\bf HDS}(X)$ both generalizes and contains the space of closed subgroups of PSL$_2{\mathbb{C}}$ with the Hausdorff-Chabauty topology, but the construction requires closer attention to fine points of general topology. In particular, since the appropriate ambient space is neither Hausdorff nor regular, the proper definition of local compactness is crucial: here it should be in the sense that every open neighborhood of a point contains a compact subneighborhood. For any topological space $X$, Fell's prescription yields a compact topological space $Fell(X)$ whose points are the closed subsets of $X$. and which is $Fell(X)$ is compact. Moreover, if $X$ is locally compact then:

(1) The space $Fell(X)$ is Hausdorff.

(2) ${F}_\eta\rightarrow{F}$ if and only if $\liminf {F}_\eta ={F}= \limsup {F}_\eta$.

(3) If $X$ is second countable then $Fell(X)$ is second countable and metrizable.

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A source of examples that comes to mind is Alexandroff topologies. For example, take the natural numbers $\mathbb{N}$, and declare a set $U \subseteq \mathbb{N}$ to be open if it is downward closed in the usual order. The closure of a point $n$ is then $\{m \in \mathbb{N}: m \geq n\}$. The poset of closed sets under reverse inclusion looks like the ordinal $\omega + 1$, and if you give this the Alexandroff topology as well (open sets = down-sets), then this will give a hypertopology.

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I hadn't seen Benjamin's comment when I was typing. –  Todd Trimble Feb 14 '13 at 11:57
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