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Alan Dow

Publications and source records attributed to Alan Dow.

At least 55 records · Page 3Linked to original sources
arXiv2018

On the tightness of $G_δ$-modifications

The $G_δ$-modification $X_δ$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_δ$ subsets of $X$. Bella and Spadaro recently investigated the connection between the values of various cardinal functions taken on $X$ and $X_δ$, respectively. In their paper, as Question 2, they raised the following problem: Is $t(X_δ) \le 2^{t(X)}$ true for every (compact) $T_2$ space $X$? Note that this is actually two questions. In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In fact, in the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the $G_δ$-modifications of countably tight, even Frechet spaces.

math.GN↗
arXiv2018

On the cofinality of the splitting number

The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)

math.LO↗
arXiv2018

Asymmetric tie-points and almost clopen subsets of N*

A tie-point of a compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. Set-theoretically a tie-point of N* is an ultrafilter whose dual maximal ideal can be generated by the union of two non-principal mod finite orthogonal ideals. We review some of the many consistency results that have depended on the construction of tie-points of N*. One especially important application, due to Velickovic, was to the existence of non-trivial involutions on N*. A tie-point of N* has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of N* in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.

math.GN↗
arXiv2016

Hereditarily normal manifolds of dimension > 1 may all be metrizable

P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercompact cardinal, and, in addition, that the manifold is hereditarily collectionwise Hausdorff. We are able to omit these extra assumptions.

math.GN↗
arXiv2016

PFA(S)[S] and countably compact spaces

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact, perfectly normal space is paracompact".

math.LO↗
arXiv2016

Normality versus paracompactness in locally compact spaces

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ω_1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω_1.

math.GN↗
arXiv2016

Reversible filters

A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-isolated point are reversible. By Stone duality, these spaces correspond to closed subsets in the Čech-Stone compactification of the natural numbers $βω$. From this, the following natural problem arises: given a space $X$ that is embeddable in $βω$, is it possible to embed $X$ in such a way that the associated filter of neighborhoods defines a reversible (or non-reversible) space? We give the solution to this problem in some cases. It is specially interesting whether the image of the required embedding is a weak $P$-set.

math.GN↗
arXiv2014

CH and the Moore-Mrowka Problem

We show that the Continuum Hypothesis is consistent with all regular spaces of hereditarily countable $π$-character being C-closed. This gives us a model of ZFC in which the Continuum Hypothesis holds and compact Hausdorff spaces of countable tightness are sequential.

math.GN↗
arXiv2014

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.

math.GN↗
arXiv2013

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.

math.GN↗
arXiv2011

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.

math.GN↗
arXiv2007

Tie-points and fixed-points in N^*

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N and in the recent study of (precisely) 2-to-1 maps on betaN setminus N . In these cases the tie-points have been the unique fixed point of an involution on betaN setminus N. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics.

math.LO↗
arXiv2007

More on Tie-points and homeomorphism in N^*

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N= N^* and in the recent study of (precisely) 2-to-1 maps on N^*. In these cases the tie-points have been the unique fixed point of an involution on N^*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of N^* which is not a homeomorph of N^* .

math.LO↗