On finite unions and finite products with the D-property

We show that the product of a subparacompact C-scattered space and a Lindelöf D-space is D. In addition, we show that every regular locally D-space which is the union of a finite collection of subparacompact spaces and metacompact spaces has the D-property. Also, we extend this result from the class of locally D-spaces to the wider class of D-scattered spaces. All the results are shown in a direct way.     

A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ₀². Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ₀². They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ₀².A modification of Alster?s Michael space that is first countable is presented.     

A note on D-spaces

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that Cp(X) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact.We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space.     

Some more examples of monotonically Lindelöf and not monotonically Lindelöf spaces

A space is monotonically Lindelöf (mL) if one can assign to every open cover U a countable open refinement r(U) (still covering the space) so that r(U) refines r(V) whenever U refines V. Some examples of mL and non-mL spaces are considered. In particular, it is shown that the product of a mL space and the convergent sequence need not be mL, that some L-spaces are mL, and that Cp(X) is mL only for countable X.     

Some results on separate and joint continuity

Let f:X×K→R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x∈X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y∈K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any ?ech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σC(X)-β-defavorable and when the assumption “base of neighborhoods” in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.     

Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

Productively Lindelöf and indestructibly Lindelöf spaces

There has recently been considerable interest in productively Lindelöf spaces, i.e. spaces such that their product with every Lindelöf space is Lindelöf. See e.g. , , , , ,  and , and work in progress by Brendle and Raghavan. Here we make several related remarks about such spaces. Indestructible Lindelöf spaces, i.e. spaces that remain Lindelöf in every countably closed forcing extension, were introduced in [28]. Their connection with topological games and selection principles was explored in [27]. We find further connections here.     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

积空间是D -空间的空间类

人们知道每个C-似空间是 D -空间, 且每个正则弱θ -可加细 C-散布空间也是D -空间。上述空间类的积空间还是D -空间吗?在这篇文章中作者讨论了该问题, 得到如下结论：正则弱θ -可加细空间的有限积是D -空间; 正则Lindel\"of C-散布空间的可数积是D -空间     

D-spaces, aD-spaces and finite unions

In this paper, we prove that if a space X is the union of a finite family of strong Σ-spaces, then X is a D-space. This gives a positive answer to a question posed by Arhangel'skii in [A.V. Arhangel'skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (2004) 2163-2170]. We also obtain results on aD-spaces and finite unions. These results improve the correspond results in [A.V. Arhangel'skii, R.Z. Buzyakova, Addition theorems and D-spaces, Comment. Math. Univ. Carolin. 43 (2002) 653-663] and [Liang-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topology Appl. 154 (2007) 469-475].     

On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ₁ is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.     

On pairwise Lindelöf bitopological spaces

In this paper, we shall continue the study of bitopological separation axioms begun by Kelly and obtained some results. Furthermore, we introduce two concepts of pairwise Lindelöf bitopological spaces and the properties for them are established. We also show that a pairwise Lindelöf space is not hereditary property.     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.     

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.     

On the compact open and finest splitting topologies

In [M.H. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145] it is shown that in the set C(Nω,N) of all continuous maps of Nω into N, where N is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since Nω is consonant (see [S. Dolecki, G.H. Greco, A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Trans. Amer. Math. Soc. 347 (1995) 2869-2884]) the Isbell topology on C(Nω,N) also is not the finest splitting topology. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property. The following spaces have the Specific Extension Property: (a) infinitely countable free unions of non-empty spaces, (b) non-compact Lindelöf zero-dimensional spaces, and (c) metric locally convex linear spaces. In particular, we prove that on the set of all real-valued functions on the (separable infinite dimensional) Hilbert space the compact-open topology does not coincide with the finest splitting topology.     

On finite unions of certain D-spaces

In this note, we show that if X is the union of a finite collection of strong Σ-spaces, then X is a D-space. As a corollary, we get a conclusion that if X is the union of a finite collection of Moore spaces, then X is a D-space. This gives a positive answer to one of Arhangel'skii's problems [A.V. Arhangel'skii, D-spaces and finite unions, Proc. AMS 132 (7) (2004) 2163-2170]. In the last part of the note, we show that if X is the union of a finite collection of DC-like spaces, then X is a D-space, where DC is the class of all discrete unions of compact spaces. As a corollary, we show that if X is the union of a finite collection of regular subparacompact C-scattered spaces, then X is a D-space.     

Monolithic spaces and D-spaces revisited

We introduce the classes of monotonically monolithic and strongly monotonically monolithic spaces. They turn out to be reasonably large and with some nice categorical properties. We prove, in particular, that any strongly monotonically monolithic countably compact space is metrizable and any monotonically monolithic space is a hereditary D-space. We show that some classes of monolithic spaces which were earlier proved to be contained in the class of D-spaces are monotonically monolithic. In particular, Cp(X) is monotonically monolithic for any Lindelöf Σ-space X. This gives a broader view of the results of Buzyakova and Gruenhage on hereditary D-property in function spaces.     

The product of two ordinals is hereditarily dually discrete

Let μ and ν be two ordinals. If X is a subspace of μ×ν, then X is dually discrete. This gives a positive answer to a question of Alas, Junqueira and Wilson. By this conclusion and a known conclusion we show that a subspace Y of μ×ν has countable spread if and only if the space Y is hereditarily a Lindelöf D-space.