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Approximation and rigidity properties in renorming constructions are characterized with some classes of simple maps. Those maps describe continuity properties up to a countable partition. The construction of such kind of maps can be done with ideas from the First Lebesgue Theorem. We present new results on the relationship between Kadec and locally uniformly rotund renormability as well as characterizations of the last one with the simple maps used here.

Continuous actions of pseudocompact groups and axioms of topological group

Alexander V. Korovin (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we show that it is possible to extend the Ellis theorem, establishing the relations between axioms of a topological group on a new class $𝒩$ of spaces containing all countably compact spaces in the case of Abelian group structure. We extend statements of the Ellis theorem concerning separate and joint continuity of group inverse on the class of spaces $𝒩$ that gives some new examples and statements for the $C_{p}$ -theory and theory of topologically homogeneous spaces.

Continuous decompositions of Peano plane continua into pseudo-arcs

Janusz Prajs (1998)

Fundamenta Mathematicae

Locally planar Peano continua admitting continuous decomposition into pseudo-arcs (into acyclic curves) are characterized as those with no local separating point. This extends the well-known result of Lewis and Walsh on a continuous decomposition of the plane into pseudo-arcs.

Continuous depencence of $δ$ on $ϵ$ as a selection property.

Costantini, Camillo, Marconi, Umberto (2005)

Mathematica Pannonica

Continuous dependence on parameters of the fixed points set for some set-valued operators

Eduard Kirr, Adrian Petruel (1997)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we extend the notion of I⁰-continuity and uniform I⁰-continuity from [2] to set-valued operators. Using these properties, we prove some results on continuous dependence of the fixed points set for families of contractive type set-valued operators.

Continuous extenders for pseudometrics

Lutzer, David J. (1978)

Seminar Uniform Spaces

Continuous extenders in normal and collectionwise normal spaces

David Lutzer, Teodor Przymusiński (1979)

Fundamenta Mathematicae

Continuous extensions: answer to a question of Hanner

Borges, C.R. (1978)

Portugaliae mathematica

Continuous extensions of multifunctions

H. A. Antosiewicz, A. Cellina (1977)

Annales Polonici Mathematici

Continuous functions between Isbell-Mrówka spaces

Salvador García-Ferreira (1998)

Commentationes Mathematicae Universitatis Carolinae

Let $Ψ (Σ)$ be the Isbell-Mr’owka space associated to the $M A D$ -family $Σ$ . We show that if $G$ is a countable subgroup of the group $𝐒 (ω)$ of all permutations of $ω$ , then there is a $M A D$ -family $Σ$ such that every $f \in G$ can be extended to an autohomeomorphism of $Ψ (Σ)$ . For a $M A D$ -family $Σ$ , we set $I n v (Σ) = {f \in 𝐒 (ω) : f [A] \in Σ$ for all $A \in Σ}$ . It is shown that for every $f \in 𝐒 (ω)$ there is a $M A D$ -family $Σ$ such that $f \in I n v (Σ)$ . As a consequence of this result we have that there is a $M A D$ -family $Σ$ such that $n + A \in Σ$ whenever $A \in Σ$ and $n < ω$ , where $n + A = {n + a : a \in A}$ for $n < ω$ . We also notice that there is no $M A D$ -family $Σ$ such...

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Continuity of Nemyckij's operator in Hölder spaces

Continuity of semigroup actions.

Continuity of the Hausdorff dimension for invariant subsets of interval maps.

Continuity of the identity embedding of Musielak-Orlicz sequence spaces

Continuity of the superposition of set-valued functions.

Continuity points in ${x} \times Y$

Continuity properties of convex-type set-valued maps.

Continuity properties up to a countable partition.