# 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)