Kuratowski convergence on compacta and Hausdorff metric convergence on compacta

Primo Brandi; Rita Ceppitelli; Ľubica Holá

Best approximations and porous sets

Simeon Reich, Alexander J. Zaslavski (2003)

Commentationes Mathematicae Universitatis Carolinae

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Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S (X)$ the family of all nonempty bounded closed convex subsets of $X$ . We endow $S (X)$ with the Hausdorff metric and show that there exists a set $ℱ \subset S (X)$ such that its complement $S (X) ∖ ℱ$ is $σ$ -porous and such that for each $A \in ℱ$ and each $\tilde{x} \in D$ , the set of solutions of the best approximation problem $∥ \tilde{x} - z ∥ \to min$ , $z \in A$ , is nonempty and compact, and each minimizing sequence has a convergent subsequence.

A $β$ -normal Tychonoff space which is not normal

Eva Murtinová (2002)

Commentationes Mathematicae Universitatis Carolinae

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$α$ -normality and $β$ -normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff $β$ -normal non-normal space and an example of a Hausdorff $α$ -normal non-regular space.

Displaying similar documents to “Kuratowski convergence on compacta and Hausdorff metric convergence on compacta”

Best approximations and porous sets

A β -normal Tychonoff space which is not normal

A $β$ -normal Tychonoff space which is not normal