Exponential topology

The weakest topology on the set of all closed subsets of a topological space in which the sets are open (in ) if is open, and closed (in ) if is closed. If , then denotes the set of all subsets of that are closed in .

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 be an arbitrary finite collection of non-empty open sets in ; a basis for the exponential topology consists of sets of the form

where denotes the point of corresponding to a given closed set . The space endowed with the exponential topology is called the exponent of the space . If is a -space, then so is . If is regular, then is a Hausdorff space. If is normal, then is completely regular. For the exponential topology normality is equivalent to compactness. If the space is compact, then so is . If is a dyadic compactum and the weight of does not exceed , then is also a dyadic compactum. On the other hand, the exponent of any compactum of weight greater than or equal to 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 . Let be a closed mapping of a space onto a space . The mapping defined by is called the exponential mapping. If is a continuous mapping of a compactum onto a compactum , then it is open if and only if the mapping is open. The functor acting from the category of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism there corresponds its exponent .

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