arXiv subjects
Publications and source records attributed to Jan van Mill.
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.
We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.
We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^ω$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $ω_1$ Cohen reals, viewed as a subspace of $2^ω$.
The pinning down number $pd(X)$ of a topological space $X$ is the smallest cardinal $κ$ such that for every neighborhood assignment $\mathcal{U}$ on $X$ there is a set of size $κ$ that meets every member of $\mathcal{U}$. Clearly, $pd(X) \le d(X)$ and we call $X$ a pd-example if $pd(X) < d(X)$. We denote by $\mathbf{S}$ the class of all singular cardinals that are not strong limit. It was proved in a paper of Juhász,Soukup and Szentmiklóssy (arXiv:1506.00206}) that TFAE: (1) $\mathbf{S} \ne \emptyset$; (2) there is a 0-dimensional $T_2$ pd-example; (3) there is a $T_2$ pd-example. The aim of this paper is to produce pd-examples with further interesting topological properties like connectivity or being a topological group by presenting several constructions that transform given pd-examples into ones with these additional properties. We show that $\mathbf{S} \ne \emptyset$ is also equivalent to the existence of a connected and locally connected $T_3$ pd-example, as well as to the existence of an abelian $T_2$ topological group pd-example. However, $\mathbf{S} \ne \emptyset$ in itself is not sufficient to imply the existence of a connected $T_{3.5}$ pd-example. But if there is $μ\in \mathbf{S}$ with $μ\ge \mathfrak{c}$ then there is an abelian $T_2$ topological group (hence $T_{3.5}$) pd-example which is also arcwise connected and locally arcwise connected. Finally, the same assumption $\,\mathbf{S} \setminus \mathfrak{c} \ne \emptyset\,$ even implies that there is a locally convex topological vector space pd-example.
Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $σ$-countably tight compactum has cardinality $\mathfrak{c}$ remains open. We also show that if an arbitrary product is $σ$-countably tight then all but finitely many of its factors must be countably tight.
We construct a homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^ω$ whose complement is dense in $2^ω$ and rigid.
We present short proofs of Toruńczyk's well-known characterization theorems of the Hilbert cube and Hilbert space, respectively.
We show that every space that is the union of a `small' family consisting of special P-sets that are F-spaces, is an F-space. We also comment on the sharpness of our results.
Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most 2n-to-1 map.
We answer a question of Juhasz by constructing under CH an example of a locally connected continuum without nontrivial convergent sequences.
We recall a characterization of hereditary indecomposability originally obtained by Krasinkiewicz and Minc, and show how it may be used to give unified constructions of various hereditarily indecomposable continua. In particular we answer a question asked by Mackowiak and Tymchatyn by showing that any continuum of arbitrary weight is a weakly confluent image of a hereditarily indecomposable continuum of the same weight. We present two methods of constructing these preimages: (a) by model-theoretic means, using the compactness and completeness theorems from first-order logic to derive these results for continua of uncountable weight from their metric counterparts; and (b) by constructing essential mappings from hereditarily indecomposable continua onto Tychonoff cubes. We finish by reviving an argument due to Kelley about hyperspaces of hereditarily indecomposable continua and show how it leads to a point-set argument that reduces Brouwer's Fixed-point theorem to its three-dimensional version.