David Lutzer

ORDERED TOPOLOGICAL SPACES

Publisher Summary This chapter presents a few topics from the theory of GO spaces. It highlights areas in which there has been recent progress and describes the way in which researchers in ordered spaces view the subject. The chapter also focuses on a LOTS or a GO space, which is a topological space already equipped with a compatible ordering. Over the years, some effort has been devoted to giving a characterization of those topological spaces for which some compatible ordering can be constructed. Results of that type are called orderability theorems. Characterizations of the arc, Cantor set, and space of irrationals might be viewed as orderability theorems. Every GO space is collectionwise normal so that, in the light of general theory, many well-known covering properties, for example sub-para-compactness, θ-refinability, and meta-compactness, are equivalent to para-compactness in a GO space.

A note on weak θ-refinability

Abstract A space is weakly θ-refinable if every open cover U of X has an open refinement V = ∪{V(n): n⩾ 1} such that given xϵX, one of the collections V(n) has finite, positive order at x. Several equivalent properties of a space are given and are used to prove that: (a) if X is weakly θ-refinable and has closed sets Gδ then X is subparacompact; (b) any quasi-developable space (in the sense of Bennett) is weekly θ-refinable; (c) a space is quasi-developable if and only if it has a θ-base, (d) a linearly ordered topological space is paracompact if and only if it is weakly θ-refinable. Examples are given which show that weak θ-refinability is strictly weaker than the notion of θ-refinability introduced by Worrell and Wicke.

Descriptive complexity of function spaces

In this paper we show that C-k(X), the set of continuous, realvalued functions on X topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in Rx even when X is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces X for which C-n(X) lies nowhere in the projective hierarchy of the complete separable metric space Rx.

Function spaces of low Borel complexity

In this paper we investigate situations in which the space C,(X) of continuous, real-valued functions on X is a Borel subset of the product space RX. We show that for completely regular, nondiscrete spaces, C,7( X) cannot be a G8, an F, or a GR, subset of RX, but it can be an Fo& or Ga,,1& subset.

A note on paracompactness in generalized ordered spaces

In this paper, we show that for generalized ordered spaces, paracompactness is equivalent to Property D, where a space X is said to have Property D if, given any collection {G(x) : x ∈ X} of open sets in X satisfying x ∈ G(x) for each x, there is a closed discrete subset D of X satisfying X = ⋃{G(x) : x ∈ D}.

A note on monotonically metacompact spaces

Abstract 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.

On dense subspaces of generalized ordered spaces

Abstract 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.

Continuous separating families in ordered spaces and strong base conditions

Abstract In this paper we study the role of Stepanovas 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.

GO-spaces with -closed discrete dense subsets

In this paper we study the question “When does a perfect generalized ordered space have a σ-closed-discrete dense subset?” and we characterize such spaces in terms of their subspace structure, s-mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a σ-closed-discrete dense set and a weak monotone ortho-base. That metrization theorem cannot be proved in ZFC for perfect GO-spaces because if there is a Souslin line, then there is a non-metrizable, perfect, linearly ordered topological space that has a weak monotone ortho-

The product of totally nonmeagre spaces

In this note we give an example of a separable, pseudo-complete metric space X which is totally nonmeagre (= every closed subspace of X is a Baire space) and yet whose square Xx X is not totally nonmeagre.