zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A natural functor for hyperspaces. (English) Zbl 1122.54006
Summary: Let $(X,{\cal T})$, $(Y,{\cal T}')$ be Hausdorff topological spaces and $CL(X)$, $CL(Y)$ respectively the families of all non empty closed subsets of $X$ and $Y$ with some hypertopologies assigned. For each function $f:X\to Y$ there is a natural function: $F:CL(X)\to CL(Y)$, where for each $A\in CL(X)$, $F(A)=cl\,f (A)\in CL(Y)$. In this paper, we study the relationship between $f$ and $F$. We are primarily interested in finding necessary and sufficient conditions on $f$ to ensure the continuity of $F$. To avoid trivial situations, we will assume that $Y$ contains at least two points and an arc and $f$ is a surjection. Since the base spaces $X,Y$ are embedded in their hyperspaces, we always have $f$ continuous. We use our recent study of Bombay topologies to get the general solution and derive results in the case of various known hypertopologies. Sample results are: (1) Let $X$ and $Y$ be metric spaces and $CL(X)$, $CL(Y)$ be assigned the corresponding Hausdorff metric topologies. Then $f$ is uniformly continuous if, and only if, $F$ is (uniformly) continuous. (2) Let $X$ and $Y$ be topological spaces and $CL(X)$, $CL(Y)$ be assigned the corresponding Vietoris topologies. Then $f$ is continuous if, and only if, $F$ is continuous. (3) Let $X$ and $Y$ be Hausdorff topological spaces and $CL(X)$, $CL(Y)$ be assigned the corresponding Fell topologies. Then $f$ is continuous and strongly compact if, and only if, $F$ is continuous.

54B20Hyperspaces (general topology)
54C05Continuous maps
54E05Proximity structures and generalizations
54E15Uniform structures and generalizations
54E35Metric spaces, metrizability