On a question of Mauldin and Ulam concerning homeomorphisms

In this paper we solve a question of Mauldin and Ulam about transformations preserving homeomorphic pairs.     

On a question by Alexey Ostrovsky concerning preservation of completeness

Let be a surjection of a zero-dimensional metrizable X onto a metrizable Y which maps clopen sets in X to locally closed (or more generally, resolvable) sets in Y. We prove that if X is completely metrizable, or hereditarily Baire, then Y has also the respective property. This strengthens some recent results of A. Ostrovsky (2009) [5] and provides an answer to his question.     

On a problem concerning generalizations of realcompact spaces

PF-compact spaces are defined. Every almost realcompact space is PF-compact and every separable PF-compact space is almost realcompact. Nonetheless in the constructible universe V = L, a space of cardinality ?₁ is PF-compact if, and only if, the closure of each of its countable subsets is almost realcompact, and ω₁ is hereditarily PF-compact.     

On a question of Maarten Maurice

In this note we give ZFC results that reduce the question of Maarten Maurice about the existence of σ-closed-discrete dense subsets of perfect generalized ordered spaces to the study of very special Baire spaces, and we discuss the current status of the question for spaces with small density. Work of Shelah, Todor?evic, Qiao, and Tall shows that Maurice's problem is undecidable for generalized ordered spaces of local density ω₁.     

On the Chatyrko-Hattori solution of de Groot's question

Let Zn=(n[0,1]×(0,1])∪(∂(n[0,1])×{0}). De Groot asked: Is cmpZn?n for every n? It is known that the answer is yes for n=1 and 2. V.A. Chatyrko and Y. Hattori [Fund. Math. 172 (2002) 107-115] showed that the answer is no for n?5. It is shown that the answer is also no for n=4. The question is unresolved for n=3.     

Solution of Shapirovskii's question

Boris Shapirovskii posed the following question: “Suppose a continuous mapping from one compact space onto another is given. Suppose that the π-character of any point in the domain is greater than the weight of the target space. Will there be two disjoint closed sets in the domain mapping onto?”. There is a zero-dimensional counterexample (Corollary 12), but the one cardinal up version is true in zero-dimensional case (Corollary 14). Nevertheless, there always will be two disjoint open sets mapping densely (Theorem 7).     

The metacompactness of spaces with bases of subinfinite rank

Let X be a set. A collection $\text{U}$of subsets of X has subinfinite rank if whenever $\text{V}$ ? $\text{U}$, ∩$\text{V}$≠ø, and $\text{V}$ is infinite, then there are two distinct elements of $\text{V}$, one of which is a subset of the other. Theorem. AT₁space with a base of subinfinite rank is hereditarily metacompact.     

A generalization of nets and bases

In this paper the notions ofK-nets andK-bases are introduced and the corresponding cardinal functions,K-netweight andK-weight, are studied. Spaces with smallK-nets orK-bases are in some sense close to compact spaces. It turns out that the ordinary weight and net-weight can be easily expressed in terms of theK-versions and some auxiliary functions, moreover under some restrictions, weight and net-weight actually coincide with the hereditary modifications ofK-weight andK-netweight, respectively.     

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.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Basic homogeneity in the class of zero-dimensional compact spaces

We show that under the continuum hypothesis there is a compact zero-dimensional space which admits a base of pairwise homeomorphic clopen subsets but it is not an h-homogeneous space (i.e. not all of its nonempty clopen subsets are homeomorphic), partially answering a question of M.V. Matveev. Under Jensen's ? principle, we can even make the space hereditarily separable and hence, by a result of Matveev, an S-space.     

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.     

On a question of Vaisman concerning complex surfaces

The last years have seen striking improvements on Vaisman’s question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is twofold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to the Vaisman’s question. We also point out a relation between lcK surfaces and generalized Kähler geometry in four-dimension and prove a new result concerning hyperbolic Inoue surfaces. We conclude with a simple observation on a question of Brunella.     

On the Urysohn number of a topological space II

Abstract

Some variations of Arhangel'skii inequality ∣X∣ = 2^χ(X)L(X) for every Hausdorff space X [3], given in [2] and [6] are improved.     

On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y₁ and Y₂ of X are said to be equivalent, if there exists a homeomorphism which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y₁ and Y₂ of X, we write Y₂?Y₁, if there exists a continuous function which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y\X is a singleton, then Y is called a one-point extension of X. Let T(X) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set (T(X),?). For a locally compact space X, we define an order-anti-isomorphism from T(X) onto the set of all nonempty closed subsets of βX\X. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point ?ech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of βX\X. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C^∗(X), partially ordered with inclusion, and the topology of subspaces of βX\X. We leave some problems open.     

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].     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

Two results on spaces with a sharp base

Example. There exists a space X with a sharp base and a perfect mapping onto a space Y which does not have a sharp base.