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Klaas Pieter Hart

Publications and source records attributed to Klaas Pieter Hart.

At least 19 recordsLinked to original sources
arXiv2026

Many subalgebras of $\mathcal{P}(ω)/\mathit{fin}$

In answer to a question on Mathoverflow we show that the Boolean algebra $\mathcal{P}(ω)/\mathit{fin}$ contains a family $\{\mathcal{B}_X:X\subseteq\mathfrak{c}\}$ of subalgebras with the property that $X\subseteq Y$ implies $\mathcal{B}_Y$ is a subalgebra of $\mathcal{B}_X$ and if $X\not\subseteq Y$ then $\mathcal{B}_Y$ is not embeddable into~$\mathcal{B}_X$. The proof proceeds by Stone duality and the construction of a suitable family of separable zero-dimensional compact spaces.

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arXiv2025

An Infinite Library

The purpose of this note is to describe a space that is regular but not completely regular, but only barely so: all closed sets are $G_δ$-sets and every singleton is a zero-set.

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arXiv2025

$\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$

Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $π: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $π^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the Čech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $π^* \circ H = h \circ π^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the Čech-Stone remainder of the half line $\mathbb H = [0,\infty)$.

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arXiv2024

Problems on $β\mathbb{N}$

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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arXiv2024

$C$-embedding, Lindelöfness, and Čech-completeness

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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arXiv2023

Closed copies of $\mathbb{N}$ in $\mathbb{R}^{ω_1}$

We investigate closed copies of~$\mathbb{N}$ in powers of~$\mathbb{R}$ with respect to $C^*$- and $C$-embedding. We show that $\mathbb{R}^{ω_1}$ contains closed copies of~$\mathbb{N}$ that are not $C^*$-embedded.

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arXiv2023

Some realcompact spaces

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 the space of natural numbers.

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arXiv2023

Long chains in the Rudin-Frolík order for uncountable cardinals

We point out that a construction by Butkovičová of a chain of length $\mathfrak{c}^+$ in the Rudin-Frolík order on $βω$ can easily be adapted to produce, given an uncountable cardinal $κ$, a chain of length $(2^κ)^+$ in the Rudin-Frolík order on $βκ$.

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arXiv2021

Universal autohomeomorphisms of $\mathbb{N}^*$

We study the existence of universal autohomeomorphisms of $\mathbb{N}^*$. We prove that $\mathsf{CH}$ implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of $\mathbb{N}^*$ are trivial.

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arXiv2021

All Parovichenko spaces may be soft-Parovichenko

It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $\mathfrak{c}$ is a remainder in a soft compactification of $\mathbb{N}$. We also exhibit an example of a compact space of weight $\aleph_1$ -- hence a remainder in some compactification of $\mathbb{N}$ -- for which it is consistent that is not the remainder in a soft compactification of $\mathbb{N}$.

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arXiv2019

Machine learning and the Continuum Hypothesis

We comment on a recent paper that connects certain forms of machine learning to Set Theory. We point out that part of the set-theoretic machinery is related to a result of Kuratowski about decompositions of finite powers of sets and we show that there is no Borel measurable monotone compression function on the unit interval.

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arXiv2018

Homogeneity and rigidity in Erdős spaces

We investigate the homogeneity of topological subspaces of separable Hilbert space, akin to the spaces with all points rational or all points irrational, so-called Erdős spaces. We provide a non-homogeneous example, that is based on one set of coordinates using, and a rigid example, based on a sequence of coordinate sets.

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arXiv2017

Brouwer and Cardinalities

This paper discusses a paper by L. E. J. Brouwer on possible cardinalities of subsets of the continuum.

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arXiv2016

Alan Dow

On the occasion of Alan Dow 60th birthday.

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