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For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Extensions of Semicontinuous Multifunctions.

Alojzy Lechicki, Sandro Levi (1990)

Forum mathematicum

$F_{σ}$ -absorbing sequences in hyperspaces of subcontinua

Helma Gladdines (1993)

Commentationes Mathematicae Universitatis Carolinae

Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟 (ℝ^{n}) = n$ for all $n$ . For a space $X$ , we denote the hyperspace consisting of all compact connected, non-empty subsets by $C (X)$ . If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${(𝒟_{\geq n} (C (X)))}_{n = 2}^{\infty}$ is $F_{σ}$ -absorbing in $C (X)$ . As a consequence, there is a homeomorphism $h : C (X) \to Q^{\infty}$ such that for all $n$ , $h [{A \in C (X) : 𝒟 (A) \geq n + 1}] = B^{n} \times Q \times Q \times \dots$ , where $B$ denotes the pseudo boundary of the Hilbert cube $Q$ . It follows that if $X$ is a countable infinite product of non-degenerate...

Fans are not c-determined

Alejandro Illanes (1999)

Colloquium Mathematicae

A continuum is a compact connected metric space. For a continuum X, let C(X) denote the hyperspace of subcontinua of X. In this paper we construct two nonhomeomorphic fans (dendroids with only one ramification point) X and Y such that C(X) and C(Y) are homeomorphic. This answers a question by Sam B. Nadler, Jr.

Fixed point property on symmetric products of chainable continua

Alejandro Illanes (2009)

Commentationes Mathematicae Universitatis Carolinae

We prove that the third symmetric product of a chainable continuum has the fixed point property.

Fixed point sets of continuum-values mappings

Jack Goodykoontz, Sam Nadler (1984)

Fundamenta Mathematicae

Function space topologies deriving from hypertopologies and networks

A. Di Concilio, A. Miranda (2001)

Bollettino dell'Unione Matematica Italiana

In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano $X$ , $Y$ spazi topologici e $C (X, Y)$ l'insieme delle funzioni continue da $X$ verso $Y$ . Fissato un «network» $α$ nel dominio $X$ ed una topologia $τ$ nell'iperspazio $C L (Y)$ del codominio $Y$ si genera una topologia $τ_{α}$ in $C (X, Y)$ richiedendo che una rete $\{f_{λ}\}$ di $C (X, Y)$ converge in $τ_{α}$ ad $f \in C (X, Y)$ se e solo se la rete $\{\bar{f_{λ} (A)}\}$ converge in $τ$ ad $\bar{f (A)}$ ...

Generalized dyadic spaces

Murray Bell (1985)

Fundamenta Mathematicae

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