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More on ordinals in topological groups

Aleksander V. Arhangel'skii, Raushan Z. Buzyakova (2008)

Commentationes Mathematicae Universitatis Carolinae

Let τ be an uncountable regular cardinal and G a T 1 topological group. We prove the following statements: (1) If τ is homeomorphic to a closed subspace of G , G is Abelian, and the order of every non-neutral element of G is greater than 5 then τ × τ embeds in G as a closed subspace. (2) If G is Abelian, algebraically generated by τ ⊂ G , and the order of every element does not exceed 3 then τ × τ is not embeddable in G . (3) There exists an Abelian topological group H such that ω 1 is homeomorphic to a closed subspace...

Moscow spaces, Pestov-Tkačenko Problem, and C -embeddings

Aleksander V. Arhangel'skii (2000)

Commentationes Mathematicae Universitatis Carolinae

We show that there exists an Abelian topological group G such that the operations in G cannot be extended to the Dieudonné completion μ G of the space G in such a way that G becomes a topological subgroup of the topological group μ G . This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

Movability and limits of polyhedra

V. Laguna, M. Moron, Nhu Nguyen, J. Sanjurjo (1993)

Fundamenta Mathematicae

We define a metric d S , called the shape metric, on the hyperspace 2 X of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace ( 2 ℝ 2 , dS) i s s e p a r a b l e . O n t h e o t h e r h a n d , w e g i v e a n e x a m p l e s h o w i n g t h a t 2ℝ2 i s n o t s e p a r a b l e i n t h e f u n d a m e n t a l m e t r i c i n t r o d u c e d b y B o r s u k .

Multifibrations. A class of shape fibrations with the path lifting property

Antonio Giraldo, José M. R. Sanjurjo (2001)

Czechoslovak Mathematical Journal

In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions,...

Multifunctions of two variables: examples and counterexamples

Jürgen Appell (1996)

Banach Center Publications

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

Multivalued fractals in b-metric spaces

Monica Boriceanu, Marius Bota, Adrian Petruşel (2010)

Open Mathematics

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal...

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