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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 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 ∈ G can be extended to an autohomeomorphism of Ψ ( Σ ) . For a M A D -family Σ , we set I n v ( Σ ) = { f ∈ 𝐒 ( ω ) : f [ A ] ∈ Σ for all A ∈ Σ } . It is shown that for every f ∈ 𝐒 ( ω ) there is a M A D -family Σ such that f ∈ 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 ∈ Σ whenever A ∈ Σ and n < ω , where n + A = { n + a : a ∈ A } for n < ω . We also notice that there is no M A D -family Σ such...

Continuous images and other topological properties of Valdivia compacta

Ondřej Kalenda (1999)

Fundamenta Mathematicae

We study topological properties of Valdivia compact spaces. We prove in particular that a compact Hausdorff space K is Corson provided each continuous image of K is a Valdivia compactum. This answers a question of M. Valdivia (1997). We also prove that the class of Valdivia compacta is stable with respect to arbitrary products and we give a generalization of the fact that Corson compacta are angelic.

Continuous images of Lindelöf p -groups, σ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

It is shown that there exists a σ -compact topological group which cannot be represented as a continuous image of a Lindelöf p -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf p -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space Y is a continuous image of a Lindelöf p -group, then there exists a covering γ of Y by dyadic compacta such that | γ | ≤ 2 ω . We also show that if a homogeneous compact space Y is a continuous...

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