Jack R. Porter

H-closed extensions I

In this chapter we begin a detailed investigation of the set H(X) of all H-closed extensions of a space X. We begin by considering strict and simple extensions of a space. We then construct and study the Fomin extension σX of an arbitrary space X, the Banaschewski-Fomin-Sanin extension μX of a semiregular space X, and one-point H-closed extensions of locally H-closed spaces. Next we consider the interrelationships among certain partitions of σX\X and the poset structure of H(X). We characterize and study those f ∈ C(X,Y) that can be extended to a function κf ∈ C(κX,κY). The chapter concludes with the study of Θ-equivalent H-closed extensions.

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.

Spaces with coarser minimal Hausdorff topologies

A technique is developed, using H-closed extensions, for deter- mining when certain Hausdorff spaces are Katetov, i.e., have a coarser minimal Hausdorff topology. Our technique works for Cech-complete Lindelof spaces, complete metrizable spaces, and many other spaces. Also, a number of inter- esting examples are presented; the most striking is an example of a Katetov space whose semiregularization is not Katetov. 1. Introduction and preliminaries. A space is called Katetov, see (BPS, G), if it has a coarser minimal Hausdorff topology. In 1941, Bourbaki (B) proved that the space Q of rational numbers does not have a coarser compact topology but could not determine whether Q is Katetov. In 1965, Herrlich (He) showed that Q is not Katetov. Since the semiregularization of an u-closed topology is a coarser minimal Haus- dorff topology, it easily follows that a space is Katetov iff it has a coarser /i-closed topology. One of the major unsolved problems in the theory of ii-closed spaces is to find an internal characterization of Katetov spaces. In this paper, tools are developed and used to show that, surprisingly, large classes of spaces are Katetov. In particular, regular, Cech-complete Lindelof spaces and complete metric spaces are Katetov; both of these results extend a result by Gryzlov (G). An example of a Katetov space is given with the most interesting property that its semiregular- ization is not Katetov. A space is shown to be Katetov if its absolute is Katetov; however, the converse is shown to be false. A countable Katetov space is shown to be scattered. If F is a compact, extremally disconnected space without isolated points and 5 is a countable discrete subspace of Y, then the complete, extremally disconnected space Y\S is shown not to be Katetov. We thank the referee for his or her comments. Now, some definitions and preliminary results needed in the sequel are presented. All spaces considered in the paper are Hausdorff. The topology of a space X is denoted by t(X), and the semiregularization of X is denoted by Xs. A coarser topology a on X, i.e., a C r(X), is sometimes called a subtopology of r(X). The basic results about u-closed and minimal Hausdorff

d-20 – H-Closed Spaces

Publisher Summary A result taught in a first course in topology is that a compact subspace of a Hausdorff space is closed. A Hausdorff space with the property of being closed in every Hausdorff space containing it as a subspace is called Hausdorff-closed (H-closed). H-closed spaces were introduced in 1924 by Alexandroff and Urysohn. They produced an example of an H-closed space that is not compact, showed that a regular H-closed space is compact, characterized a Hausdorff space as H-closed precisely when every open cover has a finite subfamily whose union is dense, and posed the question as to which Hausdorff spaces can be densely embedded in an H-closed space. These spaces enjoy many of the same properties of compact Hausdorff spaces. A compact Hausdorff space has no strictly coarser Hausdorff topology, that is, it is minimal Hausdorff. A Hausdorff space is a Katĕtov space if it has a coarser minimal Hausdorff topology. A space is Katĕtov if it is the remainder of an H-closed extension of a discrete space. This shows that every Katĕtov space is an H-set in some H-closed space.

Embedding in R-closed spaces

Abstract Some problems in the theory of R-closed spaces are solved by showing that every regular space can be embedded in a minimal regular space and there is an R-closed space with no coarser minimal regular topology. A class of spaces is found so that when fed into the Jones machinery for producing non-Tychonoff, regular spaces, the output is non-tychonoff R-closed and minimal regular spaces. Also, an example of a strongly minimal regular space that is not locally R-closed is given.

Minimal first countable spaces

In a recent paper [2], AuI I mixed the Hausdorff separation axiom withthe first countable axiom to yield a new separation axiom denoted as ffj ;a space is E\ provided every point is the countable intersection ofclosed neighborhoods. Clearly, an E\ space is I Hausdorff, and AuIproved the interesting fact that a countably compact ffj space is minimalEi (a space with a topological property P is minimal P provided thereare no strictly coarser P-topologies).In Section 2 of this paper, we derive several characterizations ofminimal ffj spaces. In particular we prove that a space is minimal E\Received 28 March 1970. This research was partly supported byUniversity of Kansas research grant No. 3^16-5038.55

Minimal extremally disconnected Hausdorff spaces

Abstract Extremally disconnected Hausdorff (abbreviated EDH) spaces that have no strictly coarser EDH topology are called minimal EDH. In this paper minimal EDH spaces are characterized in terms of the Stone-Cech compactification of such spaces. This characterization simplifies for locally compact EDH spaces X as follows: X is minimal EDH if and only if βX-X does not contain a nonempty subspace homeomorphic to some clopen subspace of X. Also, EDH-closed spaces (i.e., EDH spaces that are closed in every EDH space containing them as subspaces) are shown to be H-closed spaces. Separable minimal EDH spaces are shown to be countably compact. The problem of proving or disproving that a minimal EDH space is pseudocompact is reduced to a simpler problem and partially solved. Those Hausdorff spaces whose absolutes are minimal EDH are characterized in terms of their Fomin H-closed extension. Finally, all the EDH topologies on a space that are minimal with respect to containing a fixed regular Hausdorff topology are constructed.

Cardinal functions Fθ(X) and tθ(X) for H-closed spaces

Abstract This article extends the results proved by Cammaroto and Kočinac in 1993 by showing that tθ(X) = tθ(Xs) = t(Xs) = Fθ(X) = Fθ(Xs) = F(Xs) for every H-closed and Urysohn space. Examples are developed to study all of the relationships among the cardinal functions t, tθ, F and Fθ in the context of H-closed and Urysohn spaces. Also, an H-closed space H is constructed for which Fθ(H) < tθ(H). Two questions posed by Cammaroto and Kočinac about H-closed spaces are completely answered.

Which spaces have a coarser connected Hausdorff topology

Abstract We present some answers to the title. For example, if K is compact, zero-dimensional and D is discrete, then K⊕D has a coarser connected topology iff w(K)⩽2|D|. Similar theorems hold for ordinal spaces and spaces K⊕D where K is compact, not necessarily zero-dimensional. Every infinite cardinal has a coarser connected Hausdorff topology; so do Kunen lines, Ostaszewski spaces, and Ψ-spaces; but spaces X with X⊂βω and |βω⧹X| c do not. The statement “every locally countable, locally compact extension of ω with cardinality ω1 has a coarser connected topology” is consistent with and independent of ZFC. If X is a Hausdorff space and w(X)⩽2κ, then X can be embedded in a Hausdorff space of density κ.