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On the cardinality of power homogeneous Hausdorff spaces

G. J. Ridderbos — 2006

Fundamenta Mathematicae

We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by $d {(X)}^{π χ (X)}$ . This inequality improves many known results and it also solves a question by J. van Mill. We further introduce Δ-power homogeneity, which leads to a new proof of van Douwen’s theorem.

Group Structures and Rectifiability in Powers of Spaces

G. J. Ridderbos — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that if some power of a space X is rectifiable, then $X^{π w (X)}$ is rectifiable. It follows that no power of the Sorgenfrey line is a topological group and this answers a question of Arhangel’skiĭ. We also show that in Mal’tsev spaces of point-countable type, character and π-character coincide.

Notes on Retracts of Coset Spaces

J. van Mill; G. J. Ridderbos — 2005

Bulletin of the Polish Academy of Sciences. Mathematics

We study retracts of coset spaces. We prove that in certain spaces the set of points that are contained in a component of dimension less than or equal to n, is a closed set. Using our techniques we are able to provide new examples of homogeneous spaces that are not coset spaces. We provide an example of a compact homogeneous space which is not a coset space. We further provide an example of a compact metrizable space which is a retract of a homogeneous compact space, but which is not a retract of...

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.

Sum theorems for Ohio completeness

D. Basile; J. van Mill; G. J. Ridderbos — 2008

Colloquium Mathematicae

We present several sum theorems for Ohio completeness. We prove that Ohio completeness is preserved by taking σ-locally finite closed sums and also by taking point-finite open sums. We provide counterexamples to show that Ohio completeness is preserved neither by taking locally countable closed sums nor by taking countable open sums.

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