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Abstract.A regular space is said to be a NAC space if, given any pair of disjoint closed subsets, one of them is compact. The standard example of a noncompact NAC space is an ordinal space of uncountable cofinality. The coñnality of an arbitrary noncompact NAC space is defined, and the extent to which cofinality in NAC spaces behaves like cofinality of ordinal spaces is discussed.The notion of cofinality is fundamental in the study of ordered topological spaces. Our goal is to investigate the notion of cofinality in topological spaces. Without a linear order, we lose our sense of direction, so we study spaces where "we can approach infinity from only one direction." Even with a linear order, we must be somewhat careful; after all, every infinite discrete space has a linear order that has a cofinality, but different linear orders on a given discrete space can give different cofinalities. We want to look at analogues not of arbitrary ordered spaces, but rather those in which the cofinality can be described in terms of the topology. This idea is made precise by looking at spaces having a unique free closed ultrafilter, spaces that we call NAC spaces. We see that the theory develops smoothly for spaces that have the additional property that small subsets have compact closure, but the general theory has more questions than answers.In § 1 we give some preliminary definitions. Section 2 introduces two notions analogous to the cofinality of a linear ordered set. Several examples are given and discussed in §3. In §4 we discuss normality of products and continuous images of normal almost compact spaces. Section 5 gives other results. Preliminary definitionsAll spaces we deal with are assumed to be at least regular. A Tychonov space X is almost compact if ßX\X has at most one element. If X is almost compact but not compact, the unique element of ßX\X is denoted by oox , or if no ambiguity can result, simply oo . Examples are the space u>x of countable
Abstract.A regular space is said to be a NAC space if, given any pair of disjoint closed subsets, one of them is compact. The standard example of a noncompact NAC space is an ordinal space of uncountable cofinality. The coñnality of an arbitrary noncompact NAC space is defined, and the extent to which cofinality in NAC spaces behaves like cofinality of ordinal spaces is discussed.The notion of cofinality is fundamental in the study of ordered topological spaces. Our goal is to investigate the notion of cofinality in topological spaces. Without a linear order, we lose our sense of direction, so we study spaces where "we can approach infinity from only one direction." Even with a linear order, we must be somewhat careful; after all, every infinite discrete space has a linear order that has a cofinality, but different linear orders on a given discrete space can give different cofinalities. We want to look at analogues not of arbitrary ordered spaces, but rather those in which the cofinality can be described in terms of the topology. This idea is made precise by looking at spaces having a unique free closed ultrafilter, spaces that we call NAC spaces. We see that the theory develops smoothly for spaces that have the additional property that small subsets have compact closure, but the general theory has more questions than answers.In § 1 we give some preliminary definitions. Section 2 introduces two notions analogous to the cofinality of a linear ordered set. Several examples are given and discussed in §3. In §4 we discuss normality of products and continuous images of normal almost compact spaces. Section 5 gives other results. Preliminary definitionsAll spaces we deal with are assumed to be at least regular. A Tychonov space X is almost compact if ßX\X has at most one element. If X is almost compact but not compact, the unique element of ßX\X is denoted by oox , or if no ambiguity can result, simply oo . Examples are the space u>x of countable
It is shown that if X is a locally compact Hausdorff space, then there is a normal space Y such that ßY\Y = X. Examples are given of a countable nonsequential space, and a sequential nonlocally compact space, each of which is the Stone-Cech remainder of a normal space. A method of constructing normal almost compact spaces is presented.
Products of spaces X and Y with the property that every continuous regular image of the product is normal are investigated, and the conclusion that there are fewer such spaces than might be suspected arrived at. In fact, eitherXand Yare Lindelof, orX is countably compact, Yis compact, and t ( Y ) is small. In a first effort to find a partial converse to the second case, Nogura's result on the normality of the product of an ordinal with a compact space is extended.
PRELIMINARIESA topological space X is ACRIN if All Continuous Rcgular Images are Normal. We define 5omc tools that will help us construct nonnormal continuous regular images of normal spaces. If X is a non-Lindelof space, the cofinafity of X is cf(X) = min ((F7( : 9 is a free, countably closed filterbase of closed subsets of X ]an open cover of X with no countable subcover]. This cardinal has been investigated before under diffcrent names [4] and in different contexts [3]. The cofinality ofXis either a regular uncountable cardinal, or a singular cardinal of cofinality o. MiSEenko's example [7] is a Tychonov space with cofinality K,, but it is unknown whether there is a normal example. Since a normal example would be a Dowker space [7], countably paracompact normal spaces have regular cofinality. Supposexis a normal non-Lindeliif space with regular cofinality, witnessed by an open cover %/. We can close %under < K-unions, then inductively choose a subcover Y' = {Uu : u < K) with the property that for each CY < p < K, U,\Up,,, Up f 0. We call such "M ' increasing in type K. Given a //increasing in type K = cf(X), we pick a sequence [x, : (Y < K} such that x, E u, \ U, , , up. we say that {xu : a! < K) is a cofinakfy sequence for X and W. Note that a cofinality sequence {xu : CY < K] is right separated, and that nu
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