Monotone versions of δ-normality

According to Mack a space is countably paracompact if and only if its product with [0,1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.     

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly^? subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that,1) on weakly subparacompact spaces, countable compactness = compactness, ω₁-compactness = Lindelöfness;2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and3) on weakly subparacompact regular T₁-spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness.The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wicke?s factorization of paracompactness (over Hausdorff spaces) along with Krajewski?s are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.     

Linearly ordered covers,normality and paracompactness

It is shown that a regular space is collectionwise normal and countably paracompact if every open cover has an open, order cushioned refinement. A sufficient condition for paracompactness, in terms of certain order locally finite covers, is given, and is applied to the problem of the paracompactness of product spaces.     

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 σ-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.     

ALMOST 2-FULLY NORMAL,PAIRWISE PARACOMPACT AND COMPLETE DEVELOPABLE BISPACES

Abstract

Dedicated to Professor Sergio Salbany on the occasion of his 60th birthday.

We introduce and study the notion of an almost 2-fully normal bispace. In particular, we prove that a bispace is quasi-pseudometrizable if and only if it is almost 2-fully normal and pairwise developable. We obtain conditions under which an almost 2-fully normal bispace is subquasi-metrizable and show that the fine quasi-uniformity of any subquasi-metrizable topological space is bicomplete. We prove that every pairwise paracompact bispace (in the sense of Romaguera and Marin, 1988) is almost 2-fully normal and that the finest quasi-uniformity of any 2-Hausdorff pairwise paracompact bispace is bicomplete. We also characterize pairwise paracompactness in terms of a property of σ-Lebesgue type of the finest quasi-uniformity. Finally, we use Salbany's compactification of pairwise Tychonoff bispaces to characterize those bispaces that admit a bicomplete pair development and deduce that an interesting example of R. Fox of a non-quasi-metrizable pairwise stratifiable pairwise developable bispace admits a bicomplete pair development.     

Nagata's conjecture and countably compact hulls in generic extensions

Nagata conjectured that every M-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size ω² such that in VP Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X∈V is an M-space in VP then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in VP). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in VP. As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model VP.     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

Indivisible ultrametric spaces

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces [C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Divisibility of countable metric spaces, European J. Combin. 28 (2007) 1746-1769], we show that a countable ultrametric space is isometrically embeddable into an indivisible ultrametric space if and only if it does not contain a strictly increasing sequence of balls.     

On selections and classes of spaces

Strong paracompactness, Lindelöf number and degree of compactness are characterized in terms of selections of set-valued mappings.     

On a core concept of Arhangel'ski?

Arhangel'ski? [A.V. Arhangel'ski?, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625-634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.     

PRODUCTS IN TOPOLOGY

Abstract

Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product X^k coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.     

An extension theorem for D-normal spaces

A second countable developable T₁-space $\text{D}$₁ is defined which has the following properties: (1) $\text{D}$₁ is an absolute extensor for the class of perfect spaces. (2) $\text{D}$₁^?₀ is a universal space for second countable developable T₁-spaces.     

Erratum to “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358-362]

We correct the proof of Theorem 8 in “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358-362].     

What is a first countable space?

The definition of first countable space is standard and its meaning is very clear. But is that the case in the absence of the Axiom of Choice? The answer is negative because there are at least three choice-free versions of first countability. And, most likely, the usual definition does not correspond to what we want to be a first countable space. The three definitions as well as other characterizations of first countability are presented and it is discussed under which set-theoretic conditions they remain equivalent.     

关于遗传可遮和遗传σ-亚紧可数乘积的注记

证明了在逆序列的情形下,可遮空间、强可遮空间在假设X是可数仿紧空间的条件下可被其极限空间保持,进一步证明了遗传可遮,遗传强可遮及遗传σ-亚紧性在无需对投射及极限空间X做任何假设的情况下即可被其逆极限空间保持.作为上述两个结果的应用,分别给出了两个相关的可数Tychonoff乘积定理.     

Countable Fréchetα 1-spaces may be first countable

It is shown to be consistent that countable, Fréchet, ₁-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of length ₂.The research of both authors is partially supported by NSERC     

Different versions of a first countable space without choice

We show that two versions of a first countable topological space which are equivalent in ZFC set theory split in the absence of the Axiom of Choice AC. This answers in the negative a related question from Gutierres “What is a first countable space?”.     

On extremally disconnected topological groups

We introduce the notion of a partially selective ultrafilter and prove that (a) if G is an extremally disconnected topological group and p is a converging nonprincipal ultrafilter on G containing a countable discrete subset, then p is partially selective, and (b) the existence of a nonprincipal partially selective ultrafilter on a countable set implies the existence of a P-point in ω^∗. Thus it is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset.     

Topological classification of scattered IFS-attractors

We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal δ   we show that a countable compact space of height δ+1$\delta +1$ can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.     

Weakly continuously Urysohn spaces

We study weakly continuously Urysohn spaces, which were introduced in [P.L. Zenor, Continuously extending partial functions, Proc. Amer. Math. Soc. 135 (1) (2007) 305-312]. We show that every weakly continuously Urysohn wΔ-space has a base of countable order, that separable weakly continuously Urysohn spaces are submetrizable, hence continuously Urysohn, that monotonically normal weakly continuously Urysohn spaces are hereditarily paracompact, and that no linear extension of any uncountable subspace of the Sorgenfrey line is weakly continuously Urysohn. These results generalize various results in the literature concerning continuously Urysohn spaces.