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.     

A pseudocompactification

For a locally pseudocompact space X let

ζX=X∪clβX(βX\υX).     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

Lower and upper topologies in the Hausdorff partial order on a fixed set

In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ?σ in which there is no Hausdorff topology μ satisfying σ?μ?τ. τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies.     

On open ultrafilters and maximal points

In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1-45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter.Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195-207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157-172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216-266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime.Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T₁-topologies? Appl. Gen. Topol. 5 (2004) 231-242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619-624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187-225].     

Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F₀,F₁ of a normal T₁-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T₁-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F₀,F₁ of X admits a star-finite cover B^′ of X by members of B such that, for every B∈B^′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

On some questions about KC and related spaces

Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples:

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a KC space which cannot be embedded in any compact KC space;

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a countable KC space which does not admit any coarser compact KC topology;

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a minimal Hausdorff space which is not a k-space.

We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties.     

On extensions of topological spaces in terms of ideals

In the present paper, a kind of extension, termed ideal extension of a given topological space is considered via the concept of ideals. A general method of construction of such an extension of a T₀—space is worked out and it is finally shown that under certain condition imposed on the ideals involved, the said extension space turns out to be the compactification of a given space.     

SPACES WITH WALLMAN EQUIVALENT INTERMEDIATE SPACES

Abstract

If every infinite closed subset of the Wallman compactification, WX, of a space X must contain at least one element of X, then for any space Y intermediate between X and WX the Wallman compactification WY is homeomorphic to WX. This extends a property which characterizes normality inducing spaces. In the case where X is not normal, however, this is not a characterization, since there are nonnormal spaces for which all intermediate spaces are Wallman equivalent, but have infinite closed subsets contained in WX/X.     

Spaces with property pp

This paper continues the study of pp spaces. It is shown that under a wide variety of circumstances, pp spaces are paracompact. However, examples of pp non-paracompact spaces are given, some have strong separation and covering properties, other examples fail dramatically to be paracompact. The stability of pp is examined and contrasted with paracompactness. Various strengthenings of pp are briefly examined.     

An L space with a d-separable square

In this note I will show that an inessential modification of the ZFC example of a hereditarily Lindelöf nonseparable space which was constructed in [J.T. Moore, A solution to the L space problem, J. Amer. Math. Soc. 19 (3) (2006) 717-736] has the property that its square contains a σ-discrete dense set.     

Some localized separation axioms and their implications

In a topological spaceX, a T₂-distinct pointx means that for anyyX xy, there exist disjoint open neighbourhoods ofx andy. Similarly, T₀-distinct points and T₁distinct points are defined. In a T_i-distinct point-setA, we assume that eachxA is a T _i -distinct point (i=0, 1, 2). In the present paper some implications of these notions which localize the T _i -separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T₂-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T₀-distinct points are used to give two characterizations of the R _D -axiom.¹ In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an -limit point is stated.     

Arzelà-Ascoli theorem via the Wallman compactification

In the paper, we recall the Wallman compactification of a Tychonoff space T (denoted by Wall(T)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between C^b(T), the space of continuous and bounded functions on T , and C(Wall(T)), the space of continuous functions on the Wallman compactification of T. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-?ech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultrafilters.     

WALLMAN REMAINDERS OF REGULAR SPACES

ABSTRACT

This paper shows that the only Hausdorff spaces which can occur as Wallman remainders of Regular spaces are themselves completely regular. This is in contrast to the previously known result that any T1 space can occur as a Wallman remainder.     

Property C. Wallman basis and S-metrizability

In this present paper we prove that every Lindelof space which has a perfect locally connected Hausdorff compactification, has property C. (This latter concept was introduced by R.F. Dickman Jr). We make clear that this class of Lindelöf spaces properly contains the class of paracompact, connected, locally compact and locally connected spaces, as well as the class of those spaces whose topology can be induced by a metric with property S (or S-metrizable spaces). In this fashion, we simultaneously generalize two previous results of Dickman on spaces with property C. The use of Wallman basis with certain connectedness properties turns out to be a very convenient tool in the construction of locally connected compactifications as well as in characterizing S-metrizable spaces.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

On two selection principles and the corresponding games

We give some further results on two selection principles involving dense families of open sets and the corresponding games.     

Amalgams, connectifications, and homogeneous compacta

We construct a path-connected homogeneous compactum with cellularity c that is not homeomorphic to any product of dyadic compacta and first countable compacta. We also prove some closure properties for classes of spaces defined by various connectifiability conditions. One application is that every infinite product of infinite topological sums of Ti spaces has a Ti pathwise connectification, where i∈{1,2,3,3.5}.