R. Grant Woods

The epimorphic hull of C(X)

Abstract The epimorphic hull H(A) of a commutative semiprime ring A is defined to be the smallest von Neumann regular ring of quotients of A . Let X denote a Tychonoff space. In this paper the structure of H(C(X)) is investigated, where C(X) denotes the ring of continuous real-valued functions with domain X . Spaces X that have a regular ring of quotients of the form C(Y) are characterized, and a “minimum” such Y is found. Necessary conditions for H(C(X)) to equal C(Y) for some Y are obtained.

Subspaces of connected spaces

Abstract A connectification of a topological space X is a connected Hausdorff space that contains X as a dense subspace. Watson and Wilson have noted that a Hausdorff space with a connectification has no nonempty proper clopen H-closed subspaces. Here it is proven that a Hausdorff space in which every nonempty proper clopen set is not feebly compact and the cardinality of the set of clopen sets is at most 2 c is connectifiable. This result is used to show that every metric space with no nonempty proper clopen H-closed subspace is connectifiable, answering a question asked by Watson and Wilson. Also, there is a nonconnectifiable, Hausdorff space of cardinality c with no proper H-closed subspace. Using the set-theoretic hypothesis p = c , an example of a nonconnectifiable, normal Hausdorff space of cardinality c is constructed which has no nonempty compact open subset. This space is locally compact at all but one point, and if the continuum hypothesis is assumed it is first countable. This space provides a solution to questions asked by Watson and Wilson as well as Mack. The paper concludes by examining when extremally disconnected Tychonoff spaces have Tychonoff connectifications.

Maximal feebly compact spaces

Abstract Maximal feebly compact spaces (i.e., feebly compact spaces possessing no strictly stronger feebly compact topology) are characterized, as are special classes (countably compact, semiregular, regular) of maximal feebly compact spaces.

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 22ℵ0 pairwise disjoint, dense ℵ0-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ0-bounded.

Covering Properties and Coreflective Subcategories a, b

In 1957, Gleason2 constructed “projective covers” in the category of compact Hausdorff spaces and continuous functions by defining the “absolute” (EX, k,) of the compact Hausdorff space X. During the next ten years, this construction was generalized to other topological categories. “Projective covers” or “absolutes” of topological spaces became an important construction in categorically oriented general topology. (See reference 13 or chapter 6 of reference 8 for a detailed history.) We give a very rapid summary of the construction of absolutes. (The reader is referred to reference 8 or 13 for more details.) Associated with each T3 space X, there is an extremally disconnected zero-dimensional T3 space EX and a perfect irreducible continuous surjection k,: EX X. (A closed surjection is irreducible if proper closed subsets of its domain are mapped to proper subsets of its range.) The pair (EX, k,) is called the absolute of X, however, in an abuse of terminology, the word “absolute” sometimes refers only to the space EX. The absolute of X is unique in the sense that if j : Y X is a perfect irreducible continuous surjection and Y is an extremally disconnected zero-dimensional T3 space, then there is a homeomorphism h: Y EX such that k, o h = j . Absolutes have the following important “projective property” (e.g., see 10.51 of Walker,” or Gleason’).

Ideals of pseudocompact regular closed sets and absolutes of Hewitt realcompactifications

Abstract Let R(X) denote the Boolean algebra of regular closed subsets of the completely regular Hausdorff space X. Let E(X) denote the projective cover, or absolute, of X, and υX the Hewitt realcompactification of X. In this paper we give several characterizations of the Stone space of the factor algebra R(X)//tf(X), where I(X) denotes the ideal of pseudocompact members of R(X). As a by-product we prove that E(βX)⧹E(υX) is always a dense subspace of E(βX)⧹υE(X). Sufficient conditions are given for the Stone space of R(X)/I(X) to be homeomorphic to βN⧹\N, where N denotes the countable discrete space. Some examples are discussed.

Accumulation points of nowhere dense sets in

In this paper a ten year old problem by Kulpa and Szymanski is settled by constructing an example of a minimal Hausdorff space without isolated points which has a point that is not the accumulation point of any nowhere dense subset of the minimal Hausdorff space. Also, a result by Kulpa and Szymanski is extended by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. A decade ago, Kulpa and Szymafnski [KS] showed that each point in a compact Hausdorff space without isolated points is the accumulation point of a nowhere dense set and they asked if the result is true in the setting of minimal Hausdorff spaces without isolated points. They noted that the result is false for arbitrary H-closed spaces without isolated points. In this paper, we settle this question by giving an example of a minimal Hausdorff space without isolated points and a point which is not the accumulation point of any nowhere dense set. Also, we extend Kulpa and Szymanskis result by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. First, a few preliminary results and definitions are needed. All spaces considered in this paper are assumed to be Hausdorff. A space X is H-closed if X is closed in every space in which X is a subspace. An open set U in a space X is said to be regular open if U = int(cl U). A space X is semiregular if { U c X: U is regular open} is a base for the topology of X and is minimal Hausdorff if there is no strictly coarser Hausdorff topology on the space. Katetov [K] has shown that a space is minimal Hausdorff iff it is H-closed and semiregular. For a space X, let tX = X U {q: qi is a free open ultrafilter on X}. For each open set U c X, let oU= U U { E It X\X: int(clV) c U for some Ve 9/}. THEOREM 1. Let X be a semiregular space. The following are true: (a) [B, F, S] The family {oU: U is regular open in X} is a base for a minimal Hausdorff extension ttX of X. (b) [PW] For a regular open subset U of X, cl,,XoU = oU U cl xU. D Received by the editors May 29, 1984. 1980 Mathematics Subject Classification. Primary 54D25.

H

Abstract Let p Haus denote the category of Hausdorff spaces and p -maps, and let HCL denote the subcategory of p Haus consisting of H -closed spaces and continuous functions. It is well-known that HCL is an epireflective subcategory of p Haus. In this paper we characterize the epireflective subcategories of p Haus that contain HCL.

-closed spaces

One of the best behaved classes of functions encountered in general topology is the class of perfect functions, which was discussed in 1.8. As we have already seen, two topological spaces, one of which is the perfect continuous image of the other, will have many topological properties in common. (Examples of a number of such properties are given in 1J.) Perfect continuous surjections also play an important role in compactification theory (see 4.2(f,g) and in the study of extension properties (see 5.9(c)). Roughly speaking, if you cannot construct a homeomorphism between X and Y, constructing a perfect continuous surjection between them is the next best goal.

Epireflective subcategories of Hausdorff categories

Abstract Let P be a closed-hereditary topological property preserved by products. Call a space P -regular if it is homeomorphic to a subspace of a product of spaces with P . Suppose that each P -regular space possesses a P -regular compactification. It is well-known that each P -regular space X is densely embedded in a unique space γscPX with P such that if f: X → Y is continuous and Y has P , then f extends continuously to γscPX. Call P -pseudocompact if γscPX is compact. Associated with P is another topological property P #, possessing all the properties hypothesized for P above, defined as follows: a P -regular space X has P # if each P -pseudocompact closed subspace of X is compact. It is known that the P -pseudocompact spaces coincide with the P #-pseudocompact spaces, and that P # is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if P is not the property of being compact and P -regular, then P # is not simply generated; in other words, there does not exist a space E such that the spaces with P # are precisely those spaces homeomorphic to closed subspaces of powers of E.