arXiv subjects
Publications and source records attributed to Jan van Mill.
We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $\kappa$ there is a locally compact topological group of cardinality $2^\kappa$ that can be condensed onto a compact space but not onto a compact topological group. These answer some questions of Arhangel'skii and Buzyakova.
We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the \v{C}ech-Stone remainder of $\omega$ with the $G_\delta$-topology, is a universal object for all $P$-groups of weight at most ${\mathfrak c}$.
Answering a question raised by V. V. Tkachuk, we present several examples of $\sigma$-compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.
We show that in the class of Lindel\"of \v{C}ech-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, Lindel\"of and \v{C}ech-complete, are needed in the product result.
Sets on the boundary of a complementary component of a continuum in the plane have been of interest since the early 1920's. Curry and Mayer defined the buried points of a plane continuum to be the points in the continuum which were not on the boundary of any complementary component. Motivated by their investigations of Julia sets, they asked what happens if the set of buried points of a plane continuum is totally disconnected and non-empty. Curry, Mayer and Tymchatyn showed that in that case the continuum is Suslinian, i.e. it does not contain an uncountable collection of non-degenerate pairwise disjoint subcontinua. In an answer to a question of Curry et al, van Mill and Tuncali constructed a plane continuum whose buried point set was totally disconnected, non-empty and one-dimensional at each point of a countably infinite set. In this paper we show that the van Mill-Tuncali example was best possible in the sense that whenever the buried set is totally disconnected, then it is one-dimensional at each of at most countably many points. As a corollary we find that the buried set cannot be almost zero-dimensional unless it is zero-dimensional. We also construct locally connected van Mill-Tuncali type examples.
If $X$ is a topological space and $Y$ is any set then we call a family $\mathcal{F}$ of maps from $X$ to $Y$ nowhere constant if for every non-empty open set $U$ in $X$ there is $f \in \mathcal{F}$ with $|f[U]| > 1$, i.e. $f$ is not constant on $U$. We prove the following result that improves several earlier results in the literature. If $X$ is a topological space for which $C(X)$, the family of all continuous maps of $X$ to $\mathbb{R}$, is nowhere constant and $X$ has a $\pi$-base consisting of connected sets then $X$ is $\mathfrak{c}$-resolvable.
We show that every infinite crowded space can be mapped onto a homogeneous space of countable weight, and that there is a homogeneous space of weight continuum that cannot be mapped onto a homogeneous space of uncountable weight strictly less than continuum.
We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the \v{C}ech-Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of $\beta(\omega^2)$, thus resolving Shehtman's first problem for $n=2$. We also characterize modal logics arising from the \v{C}ech-Stone compactification of an ordinal $\gamma$ provided the Cantor normal form of $\gamma$ satisfies an additional condition. This gives a partial solution of Shehtman's second problem.
We investigate closed copies of~$\mathbb{N}$ in powers of~$\mathbb{R}$ with respect to $C^*$- and $C$-embedding. We show that $\mathbb{R}^{\omega_1}$ contains closed copies of~$\mathbb{N}$ that are not $C^*$-embedded.
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.
This is an update on, and expansion of, our paper Open problems on $\beta\omega$ in the book Open Problems in Topology.
It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space $X$ that we call the double density spectrum of $X$ and denote by $dd(X)$. We improve a result of Berner and Juhasz by showing that $dd(X)$ is always $ω$-closed (i.e. countably closed) if $X$ is Hausdorff. We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows. Let $S$ be a non-empty set of infinite cardinals. Then (1) $S = dd(X)$ holds for a Hausdorff space $X$ iff S is $ω$-closed and $sup S \le 2^{2^{\min S}},$ (2) S = dd(X) holds for a regular space X iff S is $ω$-closed and $\sup S \le {2^{\min S}}$. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If $κ= cf(κ)$ embeds in $\mathcal{P}(ω)/fin$ and $S$ is any set of uncountable regular cardinals $< κ$ with $|S| < \min S$, then there is a compactum $C$ such that $\{ω, κ\} \cup S \subset dd(C)$, moreover $λ\notin d(C)$ whenever $|S| + ω< cf(λ) < κ$ and $cf(λ) \notin S$. (ii) It is consistent to have a separable compactum $C$ such that $dd(C)$ is not $ω_1$-closed.
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.
We present some problems related to the conjugacy classes of $\mathsf{Aut}(\mathbb{N}^*)$.
We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.
We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\mathsf{CDH}$ space in ZFC.
We prove that if $X$ is a strongly locally homogeneous and locally compact separable metric space and $G$ is a region in $X$ with $\dim G=2$, then $G$ is not separated by any arc in $G$.
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.