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An invariant of bi-Lipschitz maps

Hossein Movahedi-Lankarani (1993)

Fundamenta Mathematicae

A new numerical invariant for the category of compact metric spaces and Lipschitz maps is introduced. This invariant takes a value less than or equal to 1 for compact metric spaces that are Lipschitz isomorphic to ultrametric ones. Furthermore, a theorem is provided which makes it possible to compute this invariant for a large class of spaces. In particular, by utilizing this invariant, it is shown that neither a fat Cantor set nor the set 0 ∪ 1 / n n ≥ 1 is Lipschitz isomorphic to an ultrametric space.

An observation on spaces with a zeroset diagonal

Wei-Feng Xuan (2020)

Mathematica Bohemica

We say that a space X has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. A space X has a zeroset diagonal if there is a continuous mapping f : X 2 → [ 0 , 1 ] with Δ X = f - 1 ( 0 ) , where Δ X = { ( x , x ) : x ∈ X } . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most 𝔠 .

Analytic Baire spaces

A. J. Ostaszewski (2012)

Fundamenta Mathematicae

We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.

Another note on countable Boolean algebras

Lutz Heindorf (1996)

Commentationes Mathematicae Universitatis Carolinae

We prove that a Boolean algebra is countable iff its subalgebra lattice admits a continuous complementation.

Approximate inverse systems of uniform spaces and an application of inverse systems

Michael G. Charalambous (1991)

Commentationes Mathematicae Universitatis Carolinae

The fundamental properties of approximate inverse systems of uniform spaces are established. The limit space of an approximate inverse sequence of complete metric spaces is the limit of an inverse sequence of some of these spaces. This has an application to the dimension of the limit space of an approximate inverse system. A topologically complete space with dim ≤ n is the limit of an approximate inverse system of metric polyhedra of dim ≤ n . A completely metrizable separable space with dim ≤ n is the limit of an...

Approximate quantities, hyperspaces and metric completeness

Valentín Gregori, Salvador Romaguera (2000)

Bollettino dell'Unione Matematica Italiana

Mostriamo che se X , d è uno spazio metrico completo, allora è completa anche la metrica D , indotta in modo naturale da d sul sottospazio degli insiemi sfocati («fuzzy») di X dati dalle quantità approssimate. Come è ben noto, D è una metrica molto interessante nella teoria dei punti fissi di applicazioni sfocate, poiché permette di ottenere risultati soddisfacenti in questo contesto.

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