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In this expository paper, we show how the pressing down lemma and Ulam matrices can be used to study the topology of subsets of [math] . We prove, for example, that if [math] and [math] are stationary subsets of [math] with [math] stationary, then [math] and [math] cannot be homeomorphic. Because Ulam matrices provide [math] -many pairwise disjoint stationary subsets of any given stationary set, it follows that there are [math] -many stationary subsets of any stationary subset of [math] with the property that no two of them are homeomorphic to each other. We also show that if [math] and [math] are stationary sets, then the product space [math] is normal if and only if [math] is stationary. In addition, we prove that for any [math] , [math] is normal, and that if [math] is hereditarily normal, then [math] is metrizable.

We study spaces that are continuous images of the usual space [ 0 , ω 1 ) of countable ordinals. We begin by showing that if Y is such a space and is T 3 then Y has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples show that regularity is needed in these results. We investigate when a regular continuous image of the countable ordinals must be compact, paracompact, and metrizable. For example we show that metrizability of such a Y is equivalent to each of the following: Y has a G δ -diagonal, Y is perfect, Y has a point-countable base, Y has a small diagonal in the sense of Husek, and Y has a σ-minimal base. Along the way we obtain an absolute version of the Juhasz–Szentmiklossy theorem for small spaces, proving that if Y is any compact Hausdorff space having | Y | ≤ ℵ 1 and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan.

In this paper we use Mary Ellen Rudin's solution of Nikiel's problem to investigate metrizability of certain subsets of compact monotonically normal spaces. We prove that if H is a semi-stratifiable space that can be covered by a σ-locally-finite collection of closed metrizable subspaces and if H embeds in a monotonically normal compact space, then H is metrizable. It follows that if H is a semi-stratifiable space with a monotonically normal compactification, then H is metrizable if it satisfies any one of the following: H has a σ-locally finite cover by compact subsets; H is a σ-discrete space; H is a scattered; H is σ-compact. In addition, a countable space X has a monotonically normal compactification if and only if X is metrizable. We also prove that any semi-stratifiable space with a monotonically normal compactification is first-countable and is the union of a family of dense metrizable subspaces. Having a monotonically normal compactification is a crucial hypothesis in these results because R.W. Heath has given an example of a countable non-metrizable stratifiable (and hence monotonically normal) group. We ask whether a first-countable semi-stratifiable space must be metrizable if it has a monotonically normal compactification. This is equivalent to “If X is a first-countable stratifiable space with a monotonically normal compactification, must H be metrizable?”

This paper focuses on the remarkable contributions that Mary Ellen Rudin made to the study of monotonically normal spaces.

In this paper we investigate generalized ordered (GO) spaces that have a flexible diagonal in the sense of Arhangelʼskii (2009) [2] . Spaces with a flexible diagonal are generalizations of topological groups and include spaces that are continuously homogeneous, Choban spaces, and rotoid spaces. We prove some paracompactness and metrization theorems for such spaces and construct examples of generalized ordered spaces that clarify how the types of spaces with a flexible diagonal are interrelated. We show, for example, that any GO Choban space is hereditarily paracompact, that any continuously homogeneous, first-countable GO-space is metrizable, that the space of real numbers is the only non-degenerate connected LOTS that is a Choban space, and that the Sorgenfrey line and the Michael line are Choban spaces. We extend some results of Arhangelʼskii and pose a family of questions.

In this paper we examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of -isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and mono- tonically normal spaces. We show that any monotonically normal space with property * or with a -isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces. (However, a fragmentable monotonically normal space may fail to be paracompact.) We show that any fragmentable GO-space must have a -disjoint -base and it follows from a theorem of H.E. White that any fragmentable, rst-countable GO-space has a dense metrizable subspace. We also show that any GO-space that is fragmentable and is a Baire space has a dense metrizable subspace. We show that in any compact LOTS X, metrizability is equivalent to each of the following: X is Eberlein compact; X is Talagrand compact; X is Gulko compact; X has a -isolated network; X is a Gruenhage space; X has property *; X is perfect and fragmentable; and the function space C(X) has a strictly convex dual norm. We give an example of a GO-space that has property *, is fragmentable, and has a -isolated network and a -disjoint -base but contains no dense metrizable subspace.

In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain P for which max(P) is a Gδ-subset of P and yet no measurement μ on P has ker(μ) = max(P). We also correct a mistake in the literature asserting that [0, ω1) is a space of this type. We show that if P is a Scott domain and X ⊆ max(P) is a Gδ-subset of P, then X has a Gδ-diagonal and is weakly developable. We show that if X ⊆ max(P) is a Gδ-subset of P, where P is a domain but perhaps not a Scott domain, then X is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P such that max(P) is the usual space of countable ordinals and is a Gδ-subset of P in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.

In this paper we investigate the role of domain representability and Scott-domain representability in the class of Moore spaces and the larger class of spaces with a base of countable order. We show, for example, that in a Moore space, the following are equivalent: domain representability; subcompactness; the existence of a winning strategy for player α (= the nonempty player) in the strong Choquet game Ch ( X ) ; the existence of a stationary winning strategy for player α in Ch ( X ) ; and Rudin completeness. We note that a metacompact Cech-complete Moore space described by Tall is not Scott-domain representable and also give an example of Cech-complete separable Moore space that is not co-compact and hence not Scott-domain representable. We conclude with a list of open questions.

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, Todorčevic, Qiao, and Tall shows that Maurice's problem is undecidable for generalized ordered spaces of local density ω1.

We study domain-representable spaces, i.e., spaces that can be represented as the space of maximal elements of some continuous directed-complete partial order (= domain) with the Scott topology. We show that the Michael and Sorgenfrey lines are of this type, as is any subspace of any space of ordinals. We show that any com- pletely regular space is a closed subset of some domain-representable space, and that if X is domain-representable, then so is any G�-subspace of X. It follows that any y Cech- complete space is domain-representable. These results answer several questions in the literature.

In this paper we study Banakh’s quarter-stratifiability among generalized ordered (GO)-spaces. All quarter-stratifiable GO-spaces have a σ \sigma -closed-discrete dense set and therefore are perfect, and have a G δ G_\delta -diagonal. We characterize quarter-stratifiability among GO-spaces and show that, unlike the situation in general topological spaces, quarter-stratifiability is a hereditary property in GO-spaces. We give examples showing that a separable perfect GO-space with a G δ G_\delta -diagonal can fail to be quarter-stratifiable and that any GO-space constructed on a Q-set in the real line must be quarter-stratifiable.

In this paper we characterize generalized ordered spaces that are metrizably fibered in terms of certain quotient spaces and in terms of the existence of special open covers. We apply our results to give a new characterization of perfect generalized ordered spaces that have a σ -closed-discrete dense subset and to give examples of GO-spaces that are, or are not, metrizably fibered.

In this paper we examine the interactions between the topology of certain linearly ordered topological spaces (LOTS) and the properties of trees in whose branch spaces they embed. As one example of the interaction between ordered spaces and trees, we characterize hereditary ultraparacompactness in a LOTS (or GO-space) X in terms of the possibility of embedding the space X in the branch space of a certain kind of tree.

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a Gδ-diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered space is equivalent to the existence of an OIF base and to the existence of a sharp base. We give examples showing that these are the best possible results.

In this paper we use a metric space Y Y due to A. H. Stone and one of its completions X X to construct a linearly ordered topological space E = E ( Y , X ) E = E(Y,X) that is Čech complete, has a σ \sigma -closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ \sigma -minimal base for its relative topology. However, E E is not metrizable and is not quasi-developable. The construction of E ( Y , X ) E(Y,X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone’s metric space and our construction.

For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each T ⊂ X2 −∆(X) with |T | = κ, there is an open neighborhood W of ∆(X) such that |T−W | = κ. If ω1 ∈ D(X) then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelof linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Cech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhasz and Szentmiklossy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.

In this paper we introduce a property that is a necessary and sufficient condition for a generalized ordered space X with a point-countable base to have a σ-disjoint base. The property is that there are subsets U(n) and D(n) of X such that U(n) is open in X and D(n) is a discrete-in-itself, relatively closed subset of U(n) such that if p is a point of an open set G, then for some n, we have p ϵ U(n) and D(n) ∩ G ≠ φ. This property is hereditary in a generalized ordered space X and implies hereditary paracompactness of X. We give examples to show that our results are the sharpest possible in ordered spaces and describe the role of Property III in general spaces.

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ -closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.

Publisher Summary This chapter discusses selected ordered space problems. A generalized ordered space (a GO-space) is a triple (X, Ƭ

Publisher Summary For any linearly ordered set (X

A linearly orderd topological space (addreviated LOSTS) is a triple , where is a linealy ordered set and λis the usual interval topology defined by [?]. Throughout this paper, λ,λ([?]) or λx denote the usual interval topology on a linearly ordered ...

In [5], topological spaces with various types of diagonals and weak σ-spaces are studied. In this paper thsee notions are extended and reflected upon GO-space. Recall that a LOTS (=linearly ordered topologicak space) is a topological space ...

A topological space is said to be totally paracompact (resp. totally metacompact) if every open base of it has a locally finite (resp. pointfinite) subcover. In this paper we characterize all totally paracompact GOspaces constructed on the real line. It turns out that in the class of GO-spaces on the real line, total paracompactness and total metacompactness are equivalent. Another consequence of our characterization is that totally metacompact GO-spaces on the real line are metrizable. Questions and partial results are given concerning total paracompactness in subspaces of real GO-spaces. A topological space is said to be totally paracompact [Fo] (totally metacompact) if every base of it has a locally finite subcover (point-finite subcover). R. Telgarsky and H. Kok [TK] showed that the Michael Line [M] was not totally paracompact. In [Le] A. Lelek asked if the Sorgenfrey Line [S] was totally paracompact. This question was answered negatively by J. M. O'Farrell [OF1] using a technique that showed neither the Sorgenfrey Line nor the Michael Line is totally metacompact. Since both the Sorgenfrey Line and the Michael Line are real GO-spaces the following questions naturally arise: 1. What GO-spaces on the real line are totally metacompact or even totally paracompact? 2. Is total metacompactness equivalent to total paracompactness in real GOspaces? In general the answer to Question 2 is no, since Heath's "V-space" [H] is totally metacompact and not (totally) paracompact. In this paper it will be shown that total metacompactness and total paracompactness are equivalent in real GO-spaces. Moreover, we shall completely characterize all totally paracompact real GO-spaces in Theorem 2.3. From this characterization it follows that totally paracompact real GO-spaces are metrizable. Recall that a linearly ordered topological space (=LOTS) is a linearly ordered set X equipped with the usual open interval topology. If < is the linear order on X then a subset C of X is order convex if whenever a and b are in C such that a < b, then {x E XI a < x < b} C C. A generalized ordered space (=GO-space) is a linearly ordered set equipped with a T1-topology for which there is a base consisting of convex sets. GO-spaces have been studied extensively (for example, see [BL1] or Received by the editors December 10, 1985 and, in revised form, August 14, 1986. The results in this paper were presented at the University of Southwest Louisiana Spring Topology Conference in April of 1986. This conference was sponsored in part by N.S.F. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F05, 54D18; Secondary 54E35.

The notions of an S δ set and an R δ set are introduced. A space is S -perfect ( R -perfect) if each closed set is an S δ set ( R δ ) set. Conditions are given which indicate when spaces are S -perfect or R -perfect. Examples are given of spaces which do not have these properties and examples are given of spaces with these properties that are not perfect spaces. The images of S -perfect and R -perfect spaces under various types of mappings are investigated.

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.

(1975). A Program in Mathematical Analysis for the Life and Management Sciences. The American Mathematical Monthly: Vol. 82, No. 5, pp. 514-520.

We show that every regular scattered space with a point-countable base is an open and compact image of a scattered metacompact Moore space and, consequently, is an element of the minimal class of regular spaces containing all the scattered metric spaces and invariant under open and compact mappings. This gives a characterization of a subclass of the class MOBI. Let MOBI denote the minimal class of regular spaces containing all the metric spaces and invariant under open and compact mappings (see [A]). It is easy to observe that a regular space is in MOBI if and only if it can be obtained as an image of a metric space under a mapping which is a composition of a finite number of open and compact mappings with regular domains (see [B]). The first exaxmples of nonmetacompact spaes in MOBI were constructed in [Ch]. These examples have recently been improved by a construction in [BCh] of nonweakly 0-refinable space in MOBI. A common feature of these examples is that the patheological spaces in MOBI are scattered. Let MOBI (scattered) denote the class of all the regular spaces which can be obtained as images of scattered metric spaces under mappings which are compositions of a finite number of open and compact mappings with regular domains. The aim of this note is to prove Theorem. For a regular space Y the following conditions are equivalent. (a) Y is a MOBI (scattered), (b) Y is a scattered space in MOBI, (c) Y is a scattered space with a point-countable base.

In this note, quasi-developable spaces and spaces with bases of countable order are studied. It is shown that a θ-refinable, quasi-developable space is a Moore space (and, hence, a space with a base of countable order) if it is a β-space, a wΔ-space, a p-space or an M-space. A hereditarily weak θ-refinable space with a base of countable order is a quasi-developable space. A space with a Gδ-diagonal has a base of countable order if it is a w▵-space, p-space or an M-space.

We show that any GO-space having a capacity in the sense of Scepin has a G δ-diagonal and is perfect. In addition, such a space has a σ-discrete dense subset and a dense metrizable subspace, and any GOspace having a capacity and a point-countabl e base (or having a σ-discrete dense subset and a point-countable base) is metrizable.

In this note linearly ordered topological spaces (abbreviated LOTS) with a point-countable base are examined. It is shown that a LOTS is quasi-developable if and only if it has a σ \sigma -point-finite base and a LOTS with a point-countable base is paracompact. An example of a LOTS with a point-countable base that does not have a σ \sigma -point-finite base is given. Conditions are given for the metrizability of a LOTS with a point-countable base and it is shown that a connected LOTS with a point-countable base is homeomorphic to a connected subset of the real line.

In this paper we investigate the relation between separability and the monotone Lindelof property in generalized ordered (GO)-spaces. We examine which classical examples are or are not monotonically Lindelof. Using a new technique for investigating open covers of GO-spaces, we show that any separable GO-space is hereditarily monotonically Lindelof. Finally, we investigate the relation between the hereditarily monotonically Lindelof property and the Souslin problem.

In this note we prove that every Eberlein compact linearly ordered space is metrizable. (By an Eberlein compact space we mean a topological space which can be embedded as a compact subset of a Banach space with the weak topology.)

In this paper we study spaces in which each compact subset is a G δ -set and compare them to H.W. Martin's c-semi-stratifiable (CSS) spaces, i.e. spaces in which compact sets are G δ -sets in a uniform way. We prove that a (countably) compact subset of a Hausdorff space X is metrizable and a G δ -subset of X provided X has a δθ -base, or a point-countable, T 1 -point-separating open cover, or a quasi- G δ -diagonal. We also show that any compact subset of a Hausdorff space X having a base of countable order must be a G δ -subset of X and note that this result does not hold for countably compact subsets of BCO-spaces. We characterize CSS spaces in terms of certain functions g ( n , x ) and prove a “local implies global” theorem for submetacompact spaces that are locally CSS. In addition, we give examples showing that even though every compact subset of a space with a point-countable base (respectively, of a space with a base of countable order) must be a G δ -set, there are examples of such spaces that are not CSS. In the paper's final section, we examine the role of the CSS property in the class of generalized ordered (GO) spaces. We use a stationary set argument to show that any monotonically normal CSS space is hereditarily paracompact. We show that, among GO-spaces with σ -closed-discrete dense subsets, being CSS and having a G δ -diagonal are equivalent properties, and we use a Souslin space example due to Heath to show that (consistently) the CSS property is not equivalent to the existence of a G δ -diagonal in the more general class of perfect GO-spaces.

In this paper we study four properties related to the existence of a dense metrizable subspace of a generalized ordered (GO) space. Three of the properties are classical, and one is recent. We give new characterizations of GO-spaces that have dense metrizable subspaces, investigate which GO-spaces can embed in GO-spaces with one of the four properties, and provide examples showing the relationships between the four properties.

In this paper we study the role of Stepanova's continuous separating families in the class of linearly ordered and generalized ordered spaces and we construct examples of paracompact spaces that have strong base properties (such as point-countable bases or σ -disjoint bases), have continuous separating families, and yet are non-metrizable.

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 G δ -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.

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R , then ( R , τ ) is subcompact, and so is any G δ -subspace of ( R , τ ) . We also show that if ( X , τ ) is a subcompact GO-space constructed on a subset X ⊆ R , then X is a G δ -subset of any space ( R , σ ) where σ is any GO-topology on R with τ = σ | X . It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to G δ -subsets. In addition, it follows that if ( X , τ ) is a subcompact GO-space constructed on any set of real numbers and if τ S is the topology obtained from τ by isolating all points of a set S ⊆ X , then ( X , τ S ) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known. We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R . For example, examples show that there are subcompact GO-spaces constructed on subsets X ⊆ R where X is not a G δ -subset of the usual real line. However, if ( X , τ ) is a dense-in-itself GO-space constructed on some X ⊆ R and if ( X , τ ) is subcompact (or more generally domain-representable), then ( X , τ ) contains a dense subspace Y that is a G δ -subspace of the usual real line. It follows that ( Y , τ | Y ) is a dense subcompact subspace of ( X , τ ) . Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of ( X , τ ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: “Is there a domain-representable GO-space constructed on a subset of R that is not subcompact”? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.

In this paper we study an example-machine Bush ( S , T ) where S and T are disjoint dense subsets of R . We find some topological properties that Bush ( S , T ) always has, others that it never has, and still others that Bush ( S , T ) might or might not have, depending upon the choice of the disjoint dense sets S and T . For example, we show that every Bush ( S , T ) has a point-countable base, is hereditarily paracompact, is a non-Archimedean space, is monotonically ultra-paracompact, is almost base-compact, weakly α -favorable and a Baire space, and is an α -space in the sense of Hodel. We show that Bush ( S , T ) never has a σ -relatively discrete dense subset (and therefore cannot have a dense metrizable subspace), is never Lindelof, and never has a σ -disjoint base, a σ -point-finite base, a quasi-development, a G δ -diagonal, or a base of countable order. We show that Bush ( S , T ) cannot be a β -space in the sense of Hodel and cannot be a p -space in the sense of Arhangelskii or a Σ -space in the sense of Nagami. We show that Bush ( P , Q ) is not homeomorphic to Bush ( Q , P ) . Finally, we show that a careful choice of the sets S and T can determine whether the space Bush ( S , T ) has strong completeness properties such as countable regular co-compactness, countable base compactness, countable subcompactness, and ω -Cech-completeness, and we use those results to find disjoint dense subsets S and T of R , each with cardinality 2 ω , such that Bush ( S , T ) is not homeomorphic to Bush ( T , S ) . We close with a family of questions for further study.

The panel discussed termonlogy briefly. The name "Bidirectional Reflectance Function" (BDRF) is a tongue-twister, but seems to be well established and no one has a better suggestion, so it will stick. It was pointed out that BDRF is a quantity that depends on four angles, but is usually quoted from only a few data points, so a related term such as "spread function" or "average BDRF" might be better.