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An example for mappings related to confluence

Pavel Pyrih (1999)

Archivum Mathematicum

Confluence of a mapping between topological spaces can be defined by several ways. J.J. Charatonik asked if two definitions of the confluence using the components and quasi-components are equivalent for surjective mappings with compact point inverses. We give the negative answer to this question in Example 2.1.

An example of strongly self-homeomorphic dendrite not pointwise self-homeomorphic

Pavel Pyrih (1999)

Commentationes Mathematicae Universitatis Carolinae

Such spaces in which a homeomorphic image of the whole space can be found in every open set are called self-homeomorphic. W.J. Charatonik and A. Dilks asked if any strongly self-homeomorphic dendrite is pointwise self-homeomorphic. We give a negative answer in Example 2.1.

An example related to strongly pointwise self-homeomorphic dendrites

Pavel Pyrih (1999)

Archivum Mathematicum

Such spaces in which a homeomorphic image of the whole space can be found in every open set are called self-homeomorphic. W.J. Charatonik and A. Dilks posed a problem related to strongly pointwise self-homeomorphic dendrites. We solve this problem negatively in Example 2.1.

An infinitary version of Sperner's Lemma

Aarno Hohti (2006)

Commentationes Mathematicae Universitatis Carolinae

We prove an extension of the well-known combinatorial-topological lemma of E. Sperner to the case of infinite-dimensional cubes. It is obtained as a corollary to an infinitary extension of the Lebesgue Covering Dimension Theorem.

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 irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U M : U M . Suppose X M is homeomorphic to the irrationals; must X = X M ? We have partial results. We also answer a question of Gruenhage by showing that if X M is homeomorphic to the “Long Cantor Set”, then X = X M .

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