Introduction

The concept of an H-bounded set was defined by Lambrinos in [1] as a weakening of the notion of a bounded set which he defined in [2]. H-bounded sets have arisen naturally in the author's study of extensions of Hausdorff spaces. In [3] it is shown that the notions of S- and -equivalence defined by Porter and Votaw [6] on the upper-semilattice of H-closed extensions of a Hausdorff space are equivalent if and only if every closed nowhere dense set is an H-bounded set. As a consequence, it is shown that a Hausdorff space has a unique Hausdorff extension if and only if the space is non-H-closed, almost H-closed, and every closed nowhere dense set is H-bounded. It is shown in Proposition 3.2 below that for locally H-closed (and hence almost H-closed spaces) a set is H-bounded if and only if it is the subset of an H-closed set. In [1] Lambrinos asks if this is always the case. More specifically he asks:

In light of these applications, it is important to examine H-bounded sets more throughly. The paper by Lambrinos [1] examines many basic properties of H-bounded sets. Lambrinos also examines a number of related concepts, several of which involve countable covering properties. In conjunction with these he asks:

In the second section of this paper, an example is presented which gives a negative answer to the three questions above. In the third section, H-bounded sets are examined more closely. In particular a new characterization of H-bounded sets is given and the structure of H-bounded sets is explored.

This introduction will conclude with definitions for the concepts discussed above and some basic facts. All spaces under consideration are Hausdorff spaces. In [2] Lambrinos defined a subset of a space to be bounded if every open cover of the space has a finite subcollection which covers the set. In [1] he weakened this notion is several ways. Two of these are H-bounded and countably H-bounded.

It should be noted that Lambrinos used the terminology ``almost bounded" and ``countably almost bounded" for
``H-bounded" and ``countably H-bounded". This paper will use ``H" instead of ``almost" to parallel the related terms of H-closed and H-set.

A space is H-closed if it is closed in every Hausdorff space in which it is embedded. (H-closed abbreviates Hausdorff closed.) The question of what an appropriate definition for a subset to be H-closed leads to two distinct concepts.

H-closed sets are internally defined, while H-sets are externally defined. H-bounded sets are externally defined but deal with coverings of the whole space.

Another definition due to Lambrinos is a countably H-set.

Lambrinos [1] does an excellent job of examining fundamentals of H-bounded sets and of comparing H-bounded sets,
H-sets, countably H-bounded sets, countably H-sets, and a number of other types of sets not discussed here. A few facts about these various types of sets will be mentioned for the reader's convenience and the reader is refered to [1] for a comprehensive treatment of this material.

It follows from the definitions that compact sets are
H-closed, H-closed sets are H-sets, and H-sets are H-bounded. The next example shows that the implications do not reverse.

It is easy to see that X is H-closed but not compact, is an H-set which is not H-closed, and is a closed, H-bounded set which is not an H-set.

It also follows from the definitions that every subset of a compact set, an H-closed set, or an H-set is H-bounded. Question 1.1 is asking the converse. Since every subset of a compact set is H-bounded, H-bounded sets need not be closed (which is not the case for compact sets, H-closed sets, or H-sets). Our primary interest, however, will be with closed, H-bounded sets.

Later the following filter characterizations will be needed.