S. W. Davis

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.

Cardinal functions for k-spaces

JAMES R. BOONE, SHELDON W. DAVIS AND GARY GRUENHAGEAbstract. In this paper, four cardinal functions are defined on the class offc-spaces. Some of the relationships between these cardinal functions arestudied. Characterizations of various ?-spaces are presented in terms of theexistence of these cardinal functions. A bound for the ordinal invariant k ofArhangelskii and Franklin is established in terms of the tightness of thespace. Examples are presented which exhibit the interaction between thesecardinal invariants and the ordinal invariants of Arhangelskii and Franklin.

The wΔ-space problem

Abstract We address the following question: “Must every w Δ-space with a G δ -diagonal be developable?” Consistently, the answer is “no.” Example . Assume CH. There is a zero-dimensional, scattered, locally compact, w Δ-space with a G δ -diagonal which is not developable. For normal, locally compact spaces (or slightly weaker), the answer is “yes”. Theorem . If X is ω- s CWH, locally Lindelof, w Δ-space with a G δ -diagonal, then X is developable.

Cauchy conditions on symmetrics

We call a symmetric d on a space X a wC symmetnc if whenever A C X and there exists E > 0 such that d(x, y) > E for all x, Y E A, then A is relatively discrete. We show that there are no L-spaces which admit wC symmetnrcs. The wC notion is extended to certain weaker structures such as 9t-spaces with similar results.

Compactifications of symmetrizable spaces

In response to questions of Arhangelskil, we present examples of (1) (MA + -iCH) a symmetrizable space which is not metrizable but has a completely normal compactification and (2) (CH) a symmetrizable space which is not metrizable but has a perfectly normal compactification. In the construction of (2), a technique is developed which can be used to obtain first countable compactifications of many interesting examples.

s-point finite refinable spaces

A space X is called s-point finite refinable (ds-point finite refinable) provided every open cover 𝒰 of X has an open refinement 𝒱 such that, for some (closed discrete) C⫅X,