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A continuous operator extending fuzzy ultrametrics

I. Stasyuk, Edward D. Tymchatyn (2011)

Commentationes Mathematicae Universitatis Carolinae

We consider the problem of simultaneous extension of fuzzy ultrametrics defined on closed subsets of a complete fuzzy ultrametric space. We construct an extension operator that preserves the operation of pointwise minimum of fuzzy ultrametrics with common domain and an operation which is an analogue of multiplication by a constant defined for fuzzy ultrametrics. We prove that the restriction of the extension operator onto the set of continuous, partial fuzzy ultrametrics is continuous with respect...

A continuous operator extending ultrametrics

I. Stasyuk, Edward D. Tymchatyn (2009)

Commentationes Mathematicae Universitatis Carolinae

The problem of continuous simultaneous extension of all continuous partial ultrametrics defined on closed subsets of a compact zero-dimensional metric space was recently solved by E.D. Tymchatyn and M. Zarichnyi and improvements to their result were made by I. Stasyuk. In the current paper we extend these results to complete, bounded, zero-dimensional metric spaces and to both continuous and uniformly continuous partial ultrametrics.

A note on operators extending partial ultrametrics

Edward D. Tymchatyn, Michael M. Zarichnyi (2005)

Commentationes Mathematicae Universitatis Carolinae

We consider the question of simultaneous extension of partial ultrametrics, i.e. continuous ultrametrics defined on nonempty closed subsets of a compact zero-dimensional metrizable space. The main result states that there exists a continuous extension operator that preserves the maximum operation. This extension can also be chosen so that it preserves the Assouad dimension.

A simultaneous selection theorem

Alexander D. Arvanitakis (2012)

Fundamenta Mathematicae

We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K,X) is isomorphic to C(𝓒,X) where 𝓒 denotes the Cantor set. For X = ℝ, this gives the well known Milyutin Theorem.

Bilipschitz extensions from smooth manifolds.

Taneli Huuskonen, Juha Partanen, Jussi Väisälä (1995)

Revista Matemática Iberoamericana

We prove that every compact C1-submanifold of Rn, with or without boundary, has the bilipschitz extension property in Rn.

Bootstrapping Kirszbraun's extension theorem

Eva Kopecká (2012)

Fundamenta Mathematicae

We show how Kirszbraun's theorem on extending Lipschitz mappings in Hilbert space implies its own generalization. There is a continuous extension operator preserving the Lipschitz constant of every mapping.

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