van Wouwe, J. M.

English

Institute of Mathematics AS CR, v. v. i.

Toposym 4-BJ. M. van WouweOn generalized ordered p- and M -spacesIn: Josef Novák (ed.): General topology and its relations to modern analysis and algebra IV,Proceedings of the fourth Prague topological symposium, 1976, Part B: Contributed Papers. Societyof Czechoslovak Mathematicians and Physicist, Praha, 1977. pp. [521]--525.Persistent URL: http://dml.cz/dmlcz/700640Terms of use:© Society of Czechoslovak Mathematicians and Physicist, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czOn generalized ordered p- and M-spaces. J.M. van Wouwe Amsterdam The class of all GO-spaces has "been extensively studied by a.o. D.J. Lutzer [5] and M.J. Faber [3], and most of the material in the first section can be found there. They characterized several properties, like metrizability, perfect normality and paracompactness in terms of the order-structure. We give a characterization of generalized ordered p- and M-spaces by means of the metrizability of certain quotient-spaces. The first section contains preliminary definitions and results, the second section contains the main results concerning p- and M-spaces. No elaborate proofs have been inc luded . 1. Preliminaries. Suppose (X,<) is a linearly ordered set. If p,q belong to X, then by the closed interval [p,q] we mean the set {x 6 X | p < x < q}, and by the open interval (p,q) the set {x e X | p < x < q}. Hal f-line s) like {x e X | x > p} are denoted by [p,->) etc. A subset C of X is called convex if for every pair of points p,q from C the interval [p,q] is contained in C. Every subset B of X decomposes into a collection of maximal convex subsets of X, called the convexity-components of B (in X). A point p is said to be an endpoint of a convex set C in X, if p belongs to C and C\{p} is convex. Whenever (X,<) is a linearly ordered set, the order-topology on X, with the open half-lines as subbase, is denoted by X(<) or simply by X if there is no danger of confusion. (X,<,X(<)) is called a linearly ordered topological space or LOTS. The triple (X,<,T) is called a generalized ordered topological space or GO~space if X is a set, < a linear order on it, and T a topology for X such that (i) X(<) c T (ii) T has a base consisting of convex sets. Clearly every subspace of a LOTS is a GO-space. The converse is also true: if (X,<,T) is a GO-space, and X* = {(x ,n) e X x 2 | D > 0 i f («-,x] e T \X} U u {(x ,n) e X x z [ n < 0 i f £x,-0 e T \ X } U u {(x ,n) € X x 2 | n = 0 } , 522 then X is homeomorphic to the subspace X x {0} of the LOTS (X*,-<, A(-<)), where < is the lexicographic order on X*. Let X = (X,<,T) be a GO-space. An ordered pair (A,B) of subsets of X, such that (i) X = A u B (ii) a < b for all a e A, b e B (iii) A,B e T is called a gap : if A has no right and B has no left endpoint a left pseudogap : if A (*(J>) has no right, and B has a left endpoint a right pseudogap: if A has a right, and B (*<f>) has no left endpoint. Clearly, pseudogaps do not occur in a LOTS. If (X,<,X(<)) is a LOTS, and £, = (A,B) is a gap in X, then we may regard £ ..as a "virtual element" added to X, satisfying a < £ < b for all a e A, b e B. If we add all these gaps to X, and give the resulting set the ordertopology, we obtain the Dedekind-compaotification X of X [-+]. If X = (X,<,T) is a GO-space, we think X embedded in X*, and define the Dedekind-compactification X of X as the closure of X .f in the Dedekind-compactification of X*. Then X is an ordered Hausdorff-compactifi-cation of X. It can also be obtained by "adding" to X all gaps and pseudogaps, as described above. If X and Y are topological spaces such that X c Y, then a pluming for X in Y is a sequence (U ) „ of coverings of X by open sets in Y, such that fLSt(xM ) c X n n=1 ° 11s-1 n for every x e X. A completely regular space X is a p-space if it has a pluming in its Cech-Stone-compactification, or equivalently, in any of its Hausdorff-compactifications (see [1]). Hence, a GO-space X is-a p-space iff it has a pluming in its Dedekind-compac-tification X . A space X is said to be a uAspace [2] if there exists a sequence (I/ ) * of open covers of X with property (M) below: (M) : If there exists x e X such that x^ e St(xQ,l/ ) (n = 1,2,...) then the sequence (x ) =1 has a clusterpomt. A space X is a M-space [6] if it admits a normal sequence of open covers, satisfying property (M). 2. p- and M-spaces. In the sequel, X will always denote a GO-space (X,<,T). DEFINITION If C is a convex subset of X, and £ = (A,B) is a (pseudo)gap in X, then C covers £ if C n A ;- <\> and C n B * cj>. 523 DEFINITION Every GO-space X, as a subset of X+J decomposes into convexity-compo-nents which are closed in X. Clearly the convexity-components of X in X are maximal convex subsets of X that do not cover any (pseudo)gap of X. If D is the collection of all these convexity-components 3 then the deconrpositionspace X/D is called gX3 and the identification-map is denoted by g : X ->• gX. PROPOSITION. If 6 is the identification-topology on gX3 and < is the natural order on gXj inherited from X, then gX = (gX,<j6) is a GO-space. Moreover3 the map g : X -> gX is closed and order-preserving. The main result about p-spaces is the following THEOREM X is a p-space <-> gX is metrizable. Q COROLLARY 1: X is a p-space <-> X* is a p-space. PROOF. gX is homeomorphic to g(X*). Q COROLLARY 2: Suppose X is a GO-space such that there is a (pseudo)gap between any two points of X. Then X is a p-space if and only if it is metrizable. PROOF. The fact that there is a (pseudo)gap "between any two points of X implies that gX is homeomorphic to X. D COROLLARY 3: Suppose X is a LOTS with a a-discrete dense subset. Then X is a p-space. PROOF. If D is a a-discrete, dense subset of X, then g[D], together with possible endpoints of gX, is a a-discrete, dense subset of gX, containing all y e gX such that [y,-0 or (+-9y] is an open subset of gX. Hence gX is metrizable by [3: theorem 3.1]. D Generalized ordered M-spaces can be characterized in a way similar to that of p-spaces. We need some definitions, and the following theorem. THEOREM. X is a uAspace -* X is a Vl-space. D DEFINITION. Suppose ? = (A,B) is a (pseudo)gap in X. Then £ is called countable if some strictly increasing sequence is cofinal in A, or some strictly decreasing sequence is coinitial in B. PROPOSITION. X is countably compact ** X has no countable (pseudo)gaps. D c + " . . DEFINITION. Let X be the subspace of X containing all elements of X, and all countable (pseudo)gaps of X. For every (pseudo)gap £ = (A-,Bj /from X \X add the set \ c c . . . CSJ**) of X to the sub space-topology of X if no strictly increasing sequence is cofinal in Ar, and the set («-,?] if no strictly decreasing sequence is coinitial in Bp. With the resulting topology, and relative order, X becomes a generalized ordered countably-compactification of X in the sense of Morita [7-V 524 DEFINITION. Let C be the decomposition of X into maximal convex sets that do not cover countable (pseudo)gaps3 i.e. C consists of the convexity-components of X as a subset of X . Then the decomposition-space X\C is denoted by cX and the identifica-tion map by c : X -* cX. PROPOSITION. If Y is the identification-topology on QX, and < is the natural order on cX inherited from X> then cX = (CX,<,Y) is a GO-space, and the map c : X -> cX is closed and orderpreserving. D THEOREM. The following properties are equivalent: (i) X is an M-space (ii) X has a pluming in X (Hi) cX is metrizable. D COROLLARY. X is a -p-spaoe => X is an M-space. PROOF. The map h = c°g : gX -> cX is closed, and hence preserves stratiflabi-lity. Since stratiflability is equivalent to metrizability for GO-spaces, this implies that cX is metrizable if gX is. ["] The following example shows that in the class of all GO-spaces, p-spaces and M-spa-ces do not coincide: EXAMPLE. Let 0). be the set of all countable ordinals, with the usual order, and GO* the same set with reversed order. Replace in 0). every non-limit ordinal by a copy of 03* + a)-] ,and order the resulting set X lexicographically by-<. Then the LOTS X = (X,^,A('< )) is an M-space since X has no countable gaps, so cX = {0} (in fact X is countably compact); but X is not a p-space since gX is homeomorphic to u).. and hence not metrizable. D 3. References. [1] A.V. Arhangels'kii, On a class of spaces containing all metric spaces and all locally compact spaces. Soviet. Math. Dokl. k (1963) 1051 - 1055-[2] C.J.R. Borges, On metrizability of topological spaces. Canad. J. Math. 20 (1968) 795 - 8oU. [3] M.J. Faber, Metrizability in generalized ordered spaces. Math. Centre tracts nr 53. [-+] L. Gillman and M. Henriksen, Concerning rings of continuous functions. Trans. Amer. Math. Soc. 77 (195*0 3^0-362. [5] D.J. Lutzer, On generalized ordered spaces. Dissertationes Mat. (Rozprawy Math.) 89 (1971) 1-36. 525 [6 ] K. Mor i ta , Products of normal spaces with metric spaces. Math. Ann. 15U (196-0 365 - 382. [7 ] K. Mori ta , Coiintably-compactifiable spaces. Sc i . Rep. Tokyo Daigaku Sec. A 12 (1972) 7 - 15. 

On generalized ordered $p$- and $M$-spaces

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On generalized ordered $p$ - and $M$ -spaces

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