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Removing sets from connected product spaces while preserving connectedness

Melvin Henriksen, Amir Nikou (2007)

Commentationes Mathematicae Universitatis Carolinae

As per the title, the nature of sets that can be removed from a product of more than one connected, arcwise connected, or point arcwise connected spaces while preserving the appropriate kind of connectedness is studied. This can depend on the cardinality of the set being removed or sometimes just on the cardinality of what is removed from one or two factor spaces. Sometimes it can depend on topological properties of the set being removed or its trace on various factor spaces. Some of the results...

Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, Raanan Schul (2017)

Analysis and Geometry in Metric Spaces

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...

Resolving a question of Arkhangel'skiĭ's

Michael G. Charalambous (2006)

Fundamenta Mathematicae

We construct in ZFC a cosmic space that, despite being the union of countably many metrizable subspaces, has covering dimension equal to 1 and inductive dimensions equal to 2.

Retral spaces and continua with the fixed point property

Jan van Mill, G. J. Ridderbos (2006)

Commentationes Mathematicae Universitatis Carolinae

We show that every retral continuum with the fixed point property is locally connected. It follows that an indecomposable continuum with the fixed point property is not a retract of a topological group.

Rigid extensions of ℓ -groups of continuous functions

Michelle L. Knox, Warren Wm. McGovern (2008)

Czechoslovak Mathematical Journal

Let C ( X , ℤ ) , C ( X , ℚ ) and C ( X ) denote the ℓ -groups of integer-valued, rational-valued and real-valued continuous functions on a topological space X , respectively. Characterizations are given for the extensions C ( X , ℤ ) ≤ C ( X , ℚ ) ≤ C ( X ) to be rigid, major, and dense.

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