The Katowice problem and autohomeomorphisms of $ω^*$
We show that the existence of a homeomorphism between $ω_0^*$ and $ω_1^*$ entails the existence of a non-trivial autohomeomorphism of $ω_0^*$.
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arXiv subjects
Klaas Pieter Hart
Publications and source records attributed to Klaas Pieter Hart.
On subcontinua and continuous images of beta R\R
We prove that the Cech-Stone remainder of the real line has a family of 2^c mutually non-homeomorphic subcontinua. We also exhibit a consistent example of a first-countable continuum that is not a continuous image of this remainder.
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Reflecting Lindelöf and converging omega_1-sequences
We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging omega-sequence or a non-trivial converging omega_1-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging omega_1-sequences is first-countable and, in addition, has many aleph_1-sized Lindelöf subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.
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Unions of F-spaces
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.
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A chain condition for operators from C(K)-spaces
We introduce a chain condition (bishop), defined for operators acting on C(K)-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pełczyński's characterisation of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies (bishop) if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (bishop), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying (bishop) forms a closed left ideal of B(C(K)).
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Elementary chains and compact spaces with a small diagonal
It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.
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Lelek's problem is not a metric problem
We show that Lelek's problem on the chainability of continua with span zero is not a metric problem: from a non-metric counterexample one can construct a metric one.
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A Note on Monotonically Metacompact Spaces
We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO)spaces. We show, for example, that a generalized ordered space with a sigma-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S. G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.
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Covering dimension and finite-to-one maps
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.
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A concrete co-existential map that is not confluent
We give a concrete example of a co-existential map between continua that is not confluent.
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On hereditarily indecomposable compacta and factorization of maps
We prove a general factorization theorem for maps with hereditarily indecomposable fibers and apply it to reprove a theorem of Mackoviak on the existence of universal hereditarily indecomposable continua.
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A first-countable non-remainder of H
We give a (consistent) example of a first-countable continuum that is not a remainder of the real line.
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A separable non-remainder of H
We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.
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There is no categorical metric continuum
We show there is no categorical metric continuum. This means that for every metric continuum X there is another metric continuum Y such that X and Y have (countable) elementarily equivalent bases but X and Y are not homeomorphic. As an application we show that the chainability of the pseudoarc is not a first-order property of its lattice of closed sets.
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A connected F-space
We present an example of a compact connected F-space with a continuous real-valued function f for which the union of the interiors of its fibers is not dense. This indirectly answers a question from Abramovich and Kitover in the negative.
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Elementarity and dimensions
We give an alternative proof of Fedorchuk's recent result that dim X <= Dg X for compact Hausdorff spaces X. We use the Löwenheim-Skolem theorem to reduce the problem to the metric case.
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An Algebraic and Logical approach to continuous images
Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these continuous mappings -- also tools from Model Theory. We illustrate by showing that the Čech-Stone remainder $[0,\infty)$ has a universality property akin to that of $N^*$; a theorem of Maćkowiak and Tymchatyn implies it own generalization to non-metric continua; and certain concrete compact spaces need not be continuous images of $N^*$.
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Remarks on hereditarily indecomposable continua
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.
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