Gary Gruenhage

Generalized Metric Spaces

Publisher Summary This chapter discusses generalized metric spaces. Any class of spaces defined by a property possessed by all metric spaces could be called a class of generalized metric spaces. The term is meant for classes that are close to metrizable spaces in some sense. They usually possess some of the useful properties of metric spaces, and some of the theory or techniques of metric spaces carries over to these wider classes. They can be used to characterize the images or pre-images of metric spaces under certain kinds of mappings. They often appear in theorems that characterize metrizability in terms of weaker topological properties. They should be stable under certain topological operations, such as finite or countable products, closed subspaces, and perfect mappings. This class has played an important role in the dimension theory of general spaces. The theory of generalized metric spaces is closely related to what is known as metrization theory.

Infinite games and generalizations of first-countable spaces

Abstract In this paper we introduce a new class of spaces, called W-spaces, which is defined in terms of a simple two-person infinite game. Every first-countable space is a W-space, and every W-space is countably bi-sequential. The W-space property is preserved by subspaces, Σ-products, and open mappings. Separable W-spaces are first-countable. Various other properties of W-spaces are studied, and some questions are posed.

Covering properties on X2⧹Δ, W-sets, and compact subsets of Σ-products

Abstract Let Δ ⊂ X 1 be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X 2 ⧹ Δ is metalindelof (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X , which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X .

k-spaces and products of closed images of metric spaces

We show that a recent theorem of Y. Tanaka giving necessary and sufficient conditions for the product of two closed images of metric spaces to be a ^-space is independent of the usual axioms of set theory.

Baireness of Ck(X) for locally compact X

Abstract It is an unsolved problem to characterize in terms of X when the space C k ( X ) of continuous real-valued functions on X with the compact open topology is a Baire space. In this paper, we define and study a property called the Moving Off Property, and show that for a q-space X (e.g., for X locally compact or first-countable), X has the Moving Off Property if and only if C k ( X ) is Baire.

Gσ-sets in symmetrizable and related spaces

Abstract In this paper we answer questions of Arhangelskii and Michael by providing an example of a regular symmetrizable space which is not subparacompact and has a closed subset which is not a Gσ-set. We also use the idea of sequential order to obtain some positive results, and several examples are provided which show that these results are in some sense the best possible.

Baire and Volterra spaces

In this paper we describe broad classes of spaces for which the Baire space property is equivalent to the assertion that any two dense Gδsets have dense intersection. We also provide examples of spaces where the equivalence does not hold. Finally, our techniques provide an easy proof of a new internal characterization of perfectly meager subspaces of [0, 1] and characterize metric spaces that are always of first category.

A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters

Abstract Two examples are given from the literature in which results concerning functions of a single variable are applied incorrectly to multivariate functions. One of the examples concerns the proof of a basic theorem in the theory of maximum likelihood estimation of vector-valued parameters. An alternate statement and proof of the theorem are given.

Sequential fans in topology

Abstract For an index set I , let S ( I ) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems. For example, consider the problem which ask us to compute the tightness of the square of S ( I ). We shall show that this is in fact equivalent to the well-known and more crucial topological question of W. Fleissner which asks whether, in the class of first countable spaces, the property of being collectionwise Hausdorff at certain levels implies the same property at higher levels. Next, we consider Kodamas question whether or not every Σ-product of Lasnev spaces is normal. The sequential fan again enters the scene as we show S ( ω 2 ) × S ( ω 2 ) × ω 1 , which can be embedded in a Σ-product of Lasnev spaces as a closed set, can be nonnormal in some model of set theory. On the other hand, we show that the Σ-product of arbitrarily many copies of the slightly smaller fan S ( ω 1 ) is normal.

Games, covering properties and Eberlein compacts

Abstract Consider the following game played in a locally compact space X: at the nth move, K chooses a compact set Kn⊂X, and then P chooses a point pn∉∪{Ki: i⩽n}. We say K wins ifPs points have no limit point in X. We show that X is metacompact (σ-metacompact) if and only if K has a strategy in this game which depends only on Ps last move (and the number of the move). As a corollary we obtain a game characterization of Eberlein compact spaces. We also show that if P is allowed to choose compact sets instead of points, then K has a winning strategy if and only if X is paracompact.