In this section, a new characterization of H-bounded sets is presented and the structure of H-bounded sets is examined.
The H-closed property is defined in terms of embeddings. The next theorem shows that H-bounded sets also may be characterized in terms of embeddings. The author is not aware of a similar characterization for H-sets.
Proof: 1. This follows from the definition of H-closed and the fact that an
H-closed subset is an H-closed space.
2. Necessity. Suppose X is embedded in Y. If 
let 
, and 
. 
is a free open filter
on X. It follows from Proposition 1.10 that if A is H-bounded then does
not meet A. Hence, there is a Y-neighborhood of p that misses A, and so 
. Since no points of 
are in 
it follows
that 
.
Sufficiency. Suppose, by way of contradiction,
that 
is not H-bounded. Then there is a free open filter 
on X
meeting A. Let 
be topologized as follows: A set U is open if
and 
or if 
then 
. As
is free, this topology on Y is Hausdorff. Clearly, 
and meets A
so every neighborhood of p meets A. Therefore p is in but not in
which contradicts the assumption.
Due to the usefulness of H-bounded sets in the study of extensions of Hausdorff spaces, an important problem is to understand the structure of H-bounded sets. Section 2 showed that while all subsets of H-sets are H-bounded, not all H-bounded sets are the subsets of H-sets. The next result contrasts with the example of the previous section.
Proof: Let A be an H-bounded set; then cl(A) is
also
H-bounded. Let 
is a free open filter}. Porter [5] has
shown that if X is locally H-closed then 1) 
is free and 2) is the filter
generated by { 
is H-closed}. Since cl(A) is
H-bounded, Proposition 1.10 implies that does not meet cl(A). There exists a
such that 
and 
is H-closed. This
implies that 
hence 
. Thus
A is contained in an H-closed set.
The converse to the previous proposition is not true. For example, let X be a Tychonoff space
which is not locally compact. Then is H-bounded if and only if 
is
compact.
Another type of structure problem is whether an H-bounded set can be decomposed into the union of sets with properties which are more thoroughly understood. Using the following result of Lambrinos, some progress in this direction can be made.
Proof: Let A be closed and H-bounded. Let
B=cl(int(A)). B is regular closed and a subset of A and by
Proposition 3.3 is
H-closed. Now let 
(or let 
).
H is closed nowhere dense and a subset of A, hence H-bounded (K is
nowhere dense and H-bounded).
Thus the difference between an H-bounded set and an
H-closed set is a
nowhere dense set. It is desirable to have a description of what H-bounded sets
look like. This corollary reduces the search to nowhere dense H-bounded sets.
This problem is related to the open question of characterizing the closed
subspaces of an H-closed space that are H-closed. It is interesting to compare
this corollary to the following result by Woods:
Under certain conditions an H-bounded set is an H-set. This is examined next.
Recall the notion of 
-closure. If 
then 
for all neighborhoods U of x}. A is
said to be -closed if 
. From Proposition 1.10 it
follows that a subset A of a space is H-bounded if and only if every open
filter meeting A has an adherent point in 
. The following is an
immediate corollary:
Recall that a space is Urysohn if distinct points have disjoint closed neighborhoods.
The previous two fact imply the following characterization of H-sets in terms of H-bounded sets in Urysohn spaces:
A well known result (see [8] or [7,4S(6)]) is that if every closed subspace of a space is H-closed then the space is compact. There is similar result relating H-bounded subsets to H-closed spaces.
Proof: Necessity follows from the fact that every subset of an H-closed set is H-bounded.
Conversely let 
be any open cover of nonempty sets of X. Pick 
.
is a proper closed subset and by assumption is H-bounded. There exists a finite
number of elements of whose closures cover . These together
with U form a finite subcollection of whose closures cover X.
Acknowledgements
I would like to acknowledge the great assistance of Jack Porter in this project. He originally suggested looking at Example 2.1 in conjunction with Question 1.1.