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Hausdorff topology and uniform convergence topology in spaces of continuous functions

Umberto Artico, Giuliano Marconi (1995)

Commentationes Mathematicae Universitatis Carolinae

The local coincidence of the Hausdorff topology and the uniform convergence topology on the hyperspace consisting of closed graphs of multivalued (or continuous) functions is related to the existence of continuous functions which fail to be uniformly continuous. The problem of the local coincidence of these topologies on C ( X , Y ) is investigated for some classes of spaces: topological groups, zero-dimensional spaces, metric manifolds.

Hereditarily weakly confluent induced mappings are homeomorphisms

Janusz Charatonik, Włodzimierz Charatonik (1998)

Colloquium Mathematicae

For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained for mappings between cones over the domain and over the range continua.

Homogeneity and rigidity in Erdös spaces

Klaas P. Hart, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different...

Hyperspace selections avoiding points

Valentin Gutev (2022)

Commentationes Mathematicae Universitatis Carolinae

We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...

Hyperspaces of CW-complexes

Bao-Lin Guo, Katsuro Sakai (1993)

Fundamenta Mathematicae

It is shown that the hyperspace of a connected CW-complex is an absolute retract for stratifiable spaces, where the hyperspace is the space of non-empty compact (connected) sets with the Vietoris topology.

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

By Fin(X) (resp. F i n k ( X ) ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and ℓ ₂ f ( τ ) the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if E = ℓ ₂ f ( τ ) or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes α ( τ ) and α ( τ ) of weight ≤ τ (α > 0) then Fin(E) and each F i n k ( E ) are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic...

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