K.P. Hart
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1
We show that in the class of Lindelöf Čech-complete spaces the property of being C-embedded is quite well-behaved. It admits a useful characterization that can be used to show that products and perfect preimages of C-embedded spaces are again C-embedded. We also show that both properties, Lindelöf and Čech-complete, are needed in the product result.
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We show that in the class of Lindelöf Čech-complete spaces the property of being C-embedded is quite well-behaved. It admits a useful characterization that can be used to show that products and perfect preimages of C-embedded spaces are again C-embedded. We also show that both properties, Lindelöf and Čech-complete, are needed in the product result.
We investigate closed copies of N in powers of R with respect to C⁎- and C-embedding. We show that Rω1 contains closed copies of N that are not C⁎-embedded.
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We investigate closed copies of N in powers of R with respect to C⁎- and C-embedding. We show that Rω1 contains closed copies of N that are not C⁎-embedded.
Let M=N×[0,1]. The natural projection π:M→N, which sends (n,x) to n, induces a projection mapping π⁎:M⁎→N⁎, where M⁎ and N⁎ denote the Čech-Stone remainders of M and N, respectively. We show that CH implies every autohomeomorphism of N⁎ lifts through the natural projection to an autohomeomorphism of M⁎. That is, for every homeomorphism h:N⁎→N⁎ there is a homeomorphism H:M⁎→M⁎ such that π⁎∘H=h∘π⁎. This complements a recent result of the second author, who showed that this lifting property is not a consequence of ZFC. Combining this lifting theorem with a recent result of the first author, we also prove that CH implies there is an order-reversing autohomeomorphism of H⁎, the Čech-Stone remainder of the half line H=[0,∞).
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Let M=N×[0,1]. The natural projection π:M→N, which sends (n,x) to n, induces a projection mapping π⁎:M⁎→N⁎, where M⁎ and N⁎ denote the Čech-Stone remainders of M and N, respectively. We show that CH implies every autohomeomorphism of N⁎ lifts through the natural projection to an autohomeomorphism of M⁎. That is, for every homeomorphism h:N⁎→N⁎ there is a homeomorphism H:M⁎→M⁎ such that π⁎∘H=h∘π⁎. This complements a recent result of the second author, who showed that this lifting property is not a consequence of ZFC. Combining this lifting theorem with a recent result of the first author, we also prove that CH implies there is an order-reversing autohomeomorphism of H⁎, the Čech-Stone remainder of the half line H=[0,∞).
This is an update on, and expansion of, our paper Open problems on βω in the book Open Problems in Topology.
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This is an update on, and expansion of, our paper Open problems on βω in the book Open Problems in Topology.
We present examples of realcompact spaces with closed subsets that are C*-embedded but not C-embedded, including one where the closed set is a copy of N.
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We present examples of realcompact spaces with closed subsets that are C*-embedded but not C-embedded, including one where the closed set is a copy of N.
Journal article
(2023)
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Matheus K. Bellini, Klaas Pieter Hart, Vinicius O. Rodrigues, Artur H. Tomita
We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
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We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
It is shown that, assuming the Continuum Hypothesis, every compact Hausdorff space of weight at most c is a remainder in a soft compactification of N.
We also exhibit an example of a compact space of weight aleph_1 ---hence a remainder in some compactification of N ---for which it is consistent that is not the remainder in a softcompactification of N.
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We also exhibit an example of a compact space of weight aleph_1 ---hence a remainder in some compactification of N ---for which it is consistent that is not the remainder in a softcompactification of N.
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It is shown that, assuming the Continuum Hypothesis, every compact Hausdorff space of weight at most c is a remainder in a soft compactification of N.
We also exhibit an example of a compact space of weight aleph_1 ---hence a remainder in some compactification of N ---for which it is consistent that is not the remainder in a softcompactification of N.
We also exhibit an example of a compact space of weight aleph_1 ---hence a remainder in some compactification of N ---for which it is consistent that is not the remainder in a softcompactification of N.
De verzamelingenleer ontstond aan het eind van de negentiende eeuw toen Georg Cantor kwantitatieve vragen begon te stellen over deelverzamelingen van de getallenlijn. Het negatieve antwoord op de vraag "Zijn er evenveel natuurlijke als reële getallen?" zette aan tot dieper onderzoek naar de structuur van verzamelingen en hun onderlinge verhoudingen. Dit boek laat zien hoe met verzamelingen gewerkt wordt, hoe de verzamelingenleer gebruikt kan worden om de wiskunde te grondvesten, en dat de verzamelingenleer een mooi vak is dat de moeite van het bestuderen meer dan waard is
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De verzamelingenleer ontstond aan het eind van de negentiende eeuw toen Georg Cantor kwantitatieve vragen begon te stellen over deelverzamelingen van de getallenlijn. Het negatieve antwoord op de vraag "Zijn er evenveel natuurlijke als reële getallen?" zette aan tot dieper onderzoek naar de structuur van verzamelingen en hun onderlinge verhoudingen. Dit boek laat zien hoe met verzamelingen gewerkt wordt, hoe de verzamelingenleer gebruikt kan worden om de wiskunde te grondvesten, en dat de verzamelingenleer een mooi vak is dat de moeite van het bestuderen meer dan waard is
The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging ω1-sequence.
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The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging ω1-sequence.
We present some problems related to the conjugacy classes of Aut(N^*).
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We present some problems related to the conjugacy classes of Aut(N^*).
We present an example of a zero-dimensional F-space that is not strongly zero-dimensional.
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We present an example of a zero-dimensional F-space that is not strongly zero-dimensional.
We study the existence of universal autohomeomorphisms of N
*. We prove that the Continuum Hypothesis (CH) implies there is such an auto-homeomorphism and show that there are none in any model where all auto-homeomorphisms of N
* are trivial.
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We study the existence of universal autohomeomorphisms of N *. We prove that the Continuum Hypothesis (CH) implies there is such an auto-homeomorphism and show that there are none in any model where all auto-homeomorphisms of N * are trivial.
This paper looks at an argument in 'On the vastness of Natural Languages' by D. T. Langendoen and P. M. Postal.
The conclusion is that it does not pass mathematical muster.
The salient points are
saying "language rules impose no size limits" does not mean that one can say "there are arbitrarily large entities"; it simply means that one does not avail oneself of the former assumption, the latter assumption is just that: an assumption (or better axiom), not a consequence of not using the former
the existence proof for co-ordinate projections is mathematically unsound; it establishes that "Tom and Jerry" is a sentence built from the set {Laurel, Hardy}
the proof of the main result uses the assumption "there are projections of any given cardinality"; this assumption is equivalent to the conclusion of the theorem and this reduces the main theorem to a trivial tautology ...
The conclusion is that it does not pass mathematical muster.
The salient points are
saying "language rules impose no size limits" does not mean that one can say "there are arbitrarily large entities"; it simply means that one does not avail oneself of the former assumption, the latter assumption is just that: an assumption (or better axiom), not a consequence of not using the former
the existence proof for co-ordinate projections is mathematically unsound; it establishes that "Tom and Jerry" is a sentence built from the set {Laurel, Hardy}
the proof of the main result uses the assumption "there are projections of any given cardinality"; this assumption is equivalent to the conclusion of the theorem and this reduces the main theorem to a trivial tautology ...
This paper looks at an argument in 'On the vastness of Natural Languages' by D. T. Langendoen and P. M. Postal.
The conclusion is that it does not pass mathematical muster.
The salient points are
saying "language rules impose no size limits" does not mean that one can say "there are arbitrarily large entities"; it simply means that one does not avail oneself of the former assumption, the latter assumption is just that: an assumption (or better axiom), not a consequence of not using the former
the existence proof for co-ordinate projections is mathematically unsound; it establishes that "Tom and Jerry" is a sentence built from the set {Laurel, Hardy}
the proof of the main result uses the assumption "there are projections of any given cardinality"; this assumption is equivalent to the conclusion of the theorem and this reduces the main theorem to a trivial tautology
The conclusion is that it does not pass mathematical muster.
The salient points are
saying "language rules impose no size limits" does not mean that one can say "there are arbitrarily large entities"; it simply means that one does not avail oneself of the former assumption, the latter assumption is just that: an assumption (or better axiom), not a consequence of not using the former
the existence proof for co-ordinate projections is mathematically unsound; it establishes that "Tom and Jerry" is a sentence built from the set {Laurel, Hardy}
the proof of the main result uses the assumption "there are projections of any given cardinality"; this assumption is equivalent to the conclusion of the theorem and this reduces the main theorem to a trivial tautology
This article is a reflection on the mathematical legacy of Professor Petr Simon.
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This article is a reflection on the mathematical legacy of Professor Petr Simon.
Een beschouwing naar aanleiding van het artikel 0,999 . . . = 1?
van Jos Groot, Euclides 95 (3). 18–20. Met als conclusie dat het antwoord op
de impliciete vraag natuurlijk “Ja” is.
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Een beschouwing naar aanleiding van het artikel 0,999 . . . = 1?
van Jos Groot, Euclides 95 (3). 18–20. Met als conclusie dat het antwoord op
de impliciete vraag natuurlijk “Ja” is.
The Katowice Problem is well known among topologists and set theorists. The aim of this paper is to make it known among analysts and to give Ben de Pagter something to think about in his retirement.
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The Katowice Problem is well known among topologists and set theorists. The aim of this paper is to make it known among analysts and to give Ben de Pagter something to think about in his retirement.
In January 2019 the journal Nature reported on an exciting development in Machine Learning: the very first issue of the journal Nature Machine Intelligence contains a paper that describes a learning problem whose solvability is neither provable nor refutable on the basis of the standard ZFC axioms of Set Theory. In this note K. P. Hart describes what the fuss is all about and indicates that maybe the problem is not so undecidable after all.
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In January 2019 the journal Nature reported on an exciting development in Machine Learning: the very first issue of the journal Nature Machine Intelligence contains a paper that describes a learning problem whose solvability is neither provable nor refutable on the basis of the standard ZFC axioms of Set Theory. In this note K. P. Hart describes what the fuss is all about and indicates that maybe the problem is not so undecidable after all.
The classical Erdös spaces are obtained as the subspaces of real sep- arable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different coordinates it is possible to create a rigid subspace.
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The classical Erdös spaces are obtained as the subspaces of real sep- arable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different coordinates it is possible to create a rigid subspace.