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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 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 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.

On remote points, non-normality and π -weight ω 1

Sergei Logunov (2001)

Commentationes Mathematicae Universitatis Carolinae

We show, in particular, that every remote point of X is a nonnormality point of β X if X is a locally compact Lindelöf separable space without isolated points and π w ( X ) ≤ ω 1 .

On Szymański theorem on hereditary normality of β ω

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If F is a closed subspace of ω * and the π -weight of F is countable, then every nonisolated point of F is a non-normality point of ω * . We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space" (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs...

Remainders of metrizable and close to metrizable spaces

A. V. Arhangel'skii (2013)

Fundamenta Mathematicae

We continue the study of remainders of metrizable spaces, expanding and applying results obtained in [Fund. Math. 215 (2011)]. Some new facts are established. In particular, the closure of any countable subset in the remainder of a metrizable space is a Lindelöf p-space. Hence, if a remainder of a metrizable space is separable, then this remainder is a Lindelöf p-space. If the density of a remainder Y of a metrizable space does not exceed 2 ω , then Y is a Lindelöf Σ-space. We also show that many of...

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