Darrell W. Hajek

Normality inducing spaces

Abstract It is known that every continuous function with T1 domain and T4 range has a unique Wallman extension and that every nonnormal T3 space is the range of a continuous function which is not Wallman extendible. In this paper we introduce the concept of a normality inducing space and show that if X is a T4 space, then every continuous function with domain X and T3 range is Wallman extendible if and only if X is a normality inducing space. Further if X is a normality inducing space, Y a T3 space and f : X → Y is continuous then cly(f[X] is normal.

Spaces for which all intermediate spaces are Wallman equivalent

Abstract A T 1 space X is defined to be a WSM space provided that for any space Y between X and WX , any pair of disjoint closed subsets of Y have disjoint closures in WX . The WSM spaces have the property that all spaces between X and WX have equivalent Wallman compactifications. They are also the natural generalization (from T 4 spaces to T 1 spaces) of the concept of normality inducing spaces.

SOME RESULTS CONCERNING THE WALLMAN COMPACTIFICATION

Abstract The first part of this paper surveys results and open questions about categories of T1-spaces on which the wallman compactification induces an epireflection. The second part proves results on spaces whose Wallman remainder is Hausdorff.

THE WALLMAN COMPACTIFICATION AND INTERMEDIATE SPACES

Abstract In this paper we introduce new characterizations of the Wallman compactification among the T 1 compactifications of a space. We then use one of these characterizations to investigate conditions which would imply that an intermediate Wallman space have a Wallman compactification homeomorphic to the original compactification. The most general of these conditions is that disjoint closed subsets of the intermediate space have disjoint closures in the compactification. This has, as a special case, the consequence that if X ⊆ Y ⊆ X and if Y\X is closed in WX\X, then WY ≅ W X.

Uniform continuity for all distance spaces

Abstract In this paper, we present a definition of uniform continuity which applies to morphisms in the category DST of distance spaces, and which generalizes the definitions of uniform continuity which apply in the categories of metric spaces, of nearness spaces and of zeroed distance spaces. This definition allows us to define a category BDUNIF, of d-bounded distance spaces and uniformly continuous functions (with this new definition of uniform continuity). TOP is embedded in BDUNIF as a bi-coreflective subcategory.

SPACES WITH WALLMAN EQUIVALENT INTERMEDIATE SPACES

Abstract If every infinite closed subset of the Wallman compactification, WX, of a space X must contain at least one element of X, then for any space Y intermediate between X and WX the Wallman compactification WY is homeomorphic to WX. This extends a property which characterizes normality inducing spaces. In the case where X is not normal, however, this is not a characterization, since there are nonnormal spaces for which all intermediate spaces are Wallman equivalent, but have infinite closed subsets contained in WX/X.

WALLMAN REMAINDERS OF REGULAR SPACES

ABSTRACT This paper shows that the only Hausdorff spaces which can occur as Wallman remainders of Regular spaces are themselves completely regular. This is in contrast to the previously known result that any T1 space can occur as a Wallman remainder.