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.     

A note on D-spaces and infinite unions

It is shown that if X is a countably compact space that is the union of a countable family of D-spaces, then X is compact. This gives a positive answer to Arhangel'skii's problem [A.V. Arhangel'skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (7) (2004) 2163-2170]. In this note, we also obtain a result that if a regular space X is sequential and has a point-countable k-network, then X is a D-space.     

Constructing aD, non-D-spaces

We introduce a general method to construct 0-dimensional, scattered T₂ spaces which are not linearly D. The construction is used to show that there are aD, non-D-spaces, answering a question of Arhangel?skii. The latter example is achieved using Shelah?s club guessing principles.     

D-spaces and thick covers

The following results are obtained.

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An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

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Every hereditarily thickly covered space is aD and linearly D.

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Every t-metrizable space is a D-space.

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X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

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.     

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.     

A note on D-spaces

We show that every regular T₁ submeta-Lindelöf space of cardinality ω₁ is D under MA+¬CH, which answers a question posed by Gruenhage (2011) [9]. Borges (1991) [5] asked if every monotonically normal paracompact space is a D-space, we give a characterization of paracompactness for monotonically normal spaces, which may be of some use in solving this problem.     

There are no hereditary productive γ-spaces

We show that if X is an uncountable productive γ-set [F. Jordan, Productive local properties of function spaces, Topology Appl. 154 (2007) 870-883], then there is a countable Y⊆X such that X?Y is not Hurewicz.Along the way we answer a question of A. Miller by showing that an increasing countable union of γ-spaces is again a γ-space. We will also show that λ-spaces with the Hurewicz property are precisely those spaces for which every co-countable set is Hurewicz.     

Lindelöf spaces which are indestructible, productive, or D

We discuss relationships in Lindelöf spaces among the properties “indestructible”, “productive”, “D”, and related properties.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

Universal free G-spaces

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact.     

G-compactifications of pseudocompact G-spaces

Examples of a pseudocompact (even countably compact) G-space which is not G-Tychonoff and of a locally compact pseudocompact (even countably compact) G-Tychonoff space X with βGX≠βX are constructed.     

Separate continuity, joint continuity, the Lindelöf property and p-spaces

In this paper we prove a theorem more general than the following. Suppose that X is ?ech-complete and Y is a closed subset of a product of a separable metric space with a compact Hausdorff space. Then for each separately continuous function there exists a residual set R in X such that f is jointly continuous at each point of R×Y. This confirms the suspicions of S. Mercourakis and S. Negrepontis from 1991.     

Metrization of Fpp-spaces

A Hausdorff space each subspace of which is a paracompact p-space is an F_pp-space. A space X is a closed hereditary Baire space if each closed subspace of X is a Baire space. Using a delicate theorem of Z. Balogh it is shown that a first-countable F_pp-space that is a closed hereditary Baire space is metrizable.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

A note on the rank of diagonals

For each natural number k?4, we construct a Tychonoff space with a rank k-diagonal but without a rank (k+1)-diagonal. This example proves a conjecture on rank of diagonal given by A.V. Arhangel'skii and R.Z. Buzyakova (2006) in [1] and answers some questions raised by them in the same paper.     

Urysohn universal space, its development and Hausdorff's approach

Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. More details are devoted to the unpublished Hausdorff's approach that is shown to work in a more general situation, too.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak -refinable TD-scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak -refinable -scattered space, then X is a D-space, where is the class of all D-spaces, and hence every weak -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak -refinable space then X is linearly D.