The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?     

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.     

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.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

Some new characterizations of pointfree pseudocompactness

Abstract

By [3], a frame L is pseudocompact iff every ??-sequence in L joining to the top terminates. Here it is shown, for any completely regular L, that pseudocompactness is also equivalent to (i) the analogous condition for ?-sequences, (ii) the countable almost compactness of L, (iii) the almost compactness of CozL as a σ-frame and (iv) the condition that every countably based proper filter in L clusters. Further we establish the zero-dimensional counterparts of the above, concerning the integer valued notion of pseudocompactness. Finally, we add to this a characterization of pseudocompactness in terms of uniformities.     

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.     

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.     

Local compactness and Hewitt realcompactifications of products II

We generalize and refine results from the author's paper [18]. For a completely regular Hausdorff space X, υX denotes the Hewitt realcompactification of X. It is proved that if υ(X×Y)=υX×υY for any metacompact subparacompact (or m-paracompact) space Y, then X is locally compact. A P(n)-space is a space in which every intersection of less than n open sets is open. A characterization of those spaces X such that υ (X×Y = υX×υY for any (metacompact) P(n)-space Y is also obtained.     

α-Baer rings and some related concepts via C(X)

We call a commutative ring R an F IN -ring (resp., F SA-ring) if for any two finitely generated I, J ?R we have Ann(I)+Ann(J )=Ann(I∩J ) (resp., there is K ? R such that Ann(I)+Ann(J )=Ann(K)). Moreover, we extend this concepts to αIN -rings and αSA-rings where α is a cardinal number. The class of F SA-rings includes the class of all SA-rings (hence all IN -rings) and all P P -rings (hence all Baer-rings). In this paper, after giving some properties of αSA-rings, we prove that a reduced ring R is αSA if and only if it is an αIN -ring. Consequently, C(X) is an F SA-ring if and only if C(X) is an F IN -ring and equivalently X is an F -space. Moreover, for a commutative ring R, we have shown that R is a Baer-ring if and only if R is a reduced IN -ring. A topological space X is said to be an αU E-space if the closure of any union with cardinal number less than α of clopen subsets is open. Topological properties of αU E-spaces are investigated. Finally, we show that a completely regular Hausdor? space X is an αU E-space if and only if C(X) is an αEGE-ring.     

On quasi-Nagata spaces and quasi-γ-spaces

Main results proved: (i) EveryT ₂, quasi-Nagata,-space is metrizable. (ii) Every regular,c-stratifiable, quasi-Nagata, quasi--space is metrizable.A few other theorems involvingc-semi-stratifiable spaces,W--spaces,c-Nagata spaces etc. are proved.     

Dissipated compacta

The dissipated spaces form a class of compacta which contains both the scattered compacta and the compact LOTSes (linearly ordered topological spaces), and a number of theorems true for these latter two classes are true more generally for the dissipated spaces. For example, every regular Borel measure on a dissipated space is separable.The standard Fedor?uk S-space (constructed under ?) is dissipated. A dissipated compact L-space exists iff there is a Suslin line.A product of two compact LOTSes is usually not dissipated, but it may satisfy a weakening of that property. In fact, the degree of dissipation of a space can be used to distinguish topologically a product of n LOTSes from a product of m LOTSes.     

A note onR 1-topological spaces

The purpose of the present note is to give a number of characterizations of theR ₁-axiom and to show that theR ₁-axiom is equivalent to the weakly Hausdorff axiom introduced byB. Banaschewski andJ. M. Maranda [2]. In anR ₁-space it is shown that the locally compactness property is also open hereditary and that the closure of an almost compact set is the union of the closures of its points. A necessary and sufficient condition is obtained under which a locally compact set dense in anR ₁-space is open. Finally a variant of a well-known theorem regarding two continuous functions of a topological space into aT ₂-space is formulated forR ₁-spaces.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Clean rings of continuous functions

A ring is clean if every element is the sum of a unit and an idempotent. Let be a dense local subring of the reals which is not a field. We show that the ring of A-valued continuous functions on a zero-dimensional space X is clean if and only if X is a P-space, and examine some properties of the prime ideal spectrum of this ring. Received June 3, 2005; accepted in final form December 3, 2005.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

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.