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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...

No hedgehog in the product?

Petr Simon, Gino Tironi (2002)

Commentationes Mathematicae Universitatis Carolinae

Assuming OCA, we shall prove that for some pairs of Fréchet α 4 -spaces X , Y , the Fréchetness of the product X × Y implies that X × Y is α 4 . Assuming MA, we shall construct a pair of spaces satisfying the assumptions of the theorem.

Non-additivity of the fixed point property for tree-like continua

C. L. Hagopian, M. M. Marsh (2015)

Fundamenta Mathematicae

We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge X̂ of tree-like continua, each having the fpp, and X̂ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua H and K such that H, K, and H∩ K have the fixed point property, but...

Non-normality points and nice spaces

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

J. Terasawa in " β X - { p } are non-normal for non-discrete spaces X " (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space X that each point p of its Čech–Stone remainder X * is a non-normality point of β X . We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

On butterfly-points in β X , Tychonoff products and weak Lindelöf numbers

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

Let X be the Tychonoff product ∏ α < τ X α of τ -many Tychonoff non-single point spaces X α . Let p ∈ X * be a point in the closure of some G ⊂ X whose weak Lindelöf number is strictly less than the cofinality of τ . Then we show that β X ∖ { p } is not normal. Under some additional assumptions, p is a butterfly-point in β X . In particular, this is true if either X = ω τ or X = R τ and τ is infinite and not countably cofinal.

On Countable Dense and Strong Local Homogeneity

Jan van Mill (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.

On hereditary normality of ω * , Kunen points and character ω 1

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

We show that ω * ∖ { p } is not normal, if p is a limit point of some countable subset of ω * , consisting of points of character ω 1 . Moreover, such a point p is a Kunen point and a super Kunen point.

On Mazurkiewicz sets

Marta N. Charatonik, Włodzimierz J. Charatonik (2000)

Commentationes Mathematicae Universitatis Carolinae

A Mazurkiewicz set M is a subset of a plane with the property that each straight line intersects M in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

Let a space X be Tychonoff product ∏ α < τ X α of τ -many Tychonoff nonsingle point spaces X α . Let Suslin number of X be strictly less than the cofinality of τ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification β X . In particular, this is true if X is either R τ or ω τ and a cardinal τ is infinite and not countably cofinal.

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