Exponential topology

The weakest topology on the set $ \mathop{\rm exp} X = 2 ^ {X} $ of all closed subsets of a topological space $ X $ in which the sets $ \mathop{\rm exp} A $ are open (in $ \mathop{\rm exp} X $) if $ A $ is open, and closed (in $ \mathop{\rm exp} X $) if $ A $ is closed. If $ A \subseteq X $, then $ \mathop{\rm exp} A $ denotes the set of all subsets of $ A $ that are closed in $ X $.

Example. The topology of the metric space of closed bounded subsets of a metric space endowed with the Hausdorff metric. The general definition is: Let $ U _ {1} \dots U _ {n} $ be an arbitrary finite collection of non-empty open sets in $ X $; a basis for the exponential topology consists of sets of the form

$$ \langle U _ {1} \dots U _ {n} \rangle = $$

$$ = \ \left \{ \widehat{F} \in \mathop{\rm exp} X : F \subseteq \cup _ { 1 } ^ { n } U _ {i} \& F \cap U _ {i} \neq \emptyset , i = 1 \dots n \right \} , $$

where $ \widehat{F} $ denotes the point of $ \mathop{\rm exp} X $ corresponding to a given closed set $ F \subseteq X $. The space $ \mathop{\rm exp} X $ endowed with the exponential topology is called the exponent of the space $ X $. If $ X $ is a $ T _ {1} $- space, then so is $ \mathop{\rm exp} X $. If $ X $ is regular, then $ \mathop{\rm exp} X $ is a Hausdorff space. If $ X $ is normal, then $ \mathop{\rm exp} X $ is completely regular. For the exponential topology normality is equivalent to compactness. If the space $ X $ is compact, then so is $ \mathop{\rm exp} X $. If $ X $ is a dyadic compactum and the weight of $ X $ does not exceed $ \aleph _ {1} $, then $ \mathop{\rm exp} X $ is also a dyadic compactum. On the other hand, the exponent of any compactum of weight greater than or equal to $ \aleph _ {2} $ is not a dyadic compactum. The exponent of a Peano continuum is an absolute retract in the class of metric compacta and, consequently, it is a continuous image of an interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube $ I ^ \tau $. Let $ f : X \rightarrow Y $ be a closed mapping of a space $ X $ onto a space $ T $. The mapping $ \mathop{\rm exp} f : \mathop{\rm exp} X \rightarrow \mathop{\rm exp} Y $ defined by $ ( \mathop{\rm exp} f ) ( \widehat{F} ) = ( f ( F) ) \widehat{ {}} $ is called the exponential mapping. If $ f : X \rightarrow Y $ is a continuous mapping of a compactum $ X $ onto a compactum $ Y $, then it is open if and only if the mapping $ \mathop{\rm exp} f $ is open. The functor $ \mathop{\rm exp} X $ acting from the category of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism $ f $ there corresponds its exponent $ \mathop{\rm exp} f $.

References

[1]	K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French)

Comments

The exponential topology is better known as the Vietoris topology and the exponent of a space is usually called its hyperspace, cf. also Hyperspace. Concerning hyperspaces of Peano continua, it was shown in [a1] that these are in fact homeomorphic to the Hilbert cube.

References

[a1]	D.W. Curtis, A.M. Schori, "Hyperspaces of Peano continua are Hilbert cubes" Fund. Math. , 101 (1978) pp. 19–38