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The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces

M. Holmes (1992)

Fundamenta Mathematicae

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry....

Topological spaces admitting a unique fractal structure

Christoph Bandt, T. Retta (1992)

Fundamenta Mathematicae

Each homeomorphism from the n-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, "fractal structure" is not a metric but a topological phenomenon.

Totally bounded frame quasi-uniformities

Peter Fletcher, Worthen N. Hunsaker, William F. Lindgren (1993)

Commentationes Mathematicae Universitatis Carolinae

This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity ◃ on a frame L there is a totally bounded quasi-uniformity on L that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines ◃ . The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψ L and the compactification ℜ L of a uniform frame ( L , 𝐔 ) are meaningful for quasi-uniform frames. If 𝐔 is a totally bounded quasi-uniformity...

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