Displaying similar documents to “Preservation of properties of a map by forcing”

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

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

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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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 the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d > 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets X , Y and a function f : X Y which is onto and n -to-one for some n , the preimage of any ultrafilter of Y under f extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms A and a finite-to-one onto function f : A ω such that for each free ultrafilter of ω its preimage under f does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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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.

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

On the solvability of systems of linear equations over the ring of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ( E i ) i I of homogeneous linear equations over is non-trivially solvable in provided that each subsystem ( E j ) j J with | J | c is non-trivially solvable in where c is a fixed cardinal number such that c < | I | . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., I being finite). (b) The answer is ‘No’ in the denumerable case (i.e., | I | = 0 and c a natural number). (c) The answer in case that I is uncountable and c 0 is ‘No...

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

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CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong n 1 -collection...........................................................