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
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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 |
Exponential topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exponential_topology&oldid=14284