An order theoretic characterization of nested spaces
Partial orders in topological spaces have been of great interest to topologists since the classic work of Nachbin [7]. Later L. E. Ward, Jr. [11] studied partially ordered topological spaces and in 1954 gave a characterization of trees in terms of an inherent partial order [12]. Recently Fugate, Gordh and Lum [1] and [2] gave an order theoretic characterization of arc smooth continua. Other similar results were obtained by Smithson [8], Smithson and Muenzenberger [4], and Ward [13]. Another area in which partial orders have played a key role is the study of dendritic spaces. See, in particular, the comprehensive work of Ward [14] and the recent results of Muenzenberger and Smithson [5] and [6]. In 1946, G. S. Young, Jr. [15] introduced the notion of a nested space. Nested spaces possess a natural order structure which has proven useful in obtaining fixed point theorems (see, for example, [9] and [10]). The algebraic properties of nested spaces were studied extensively by Muenzenberger and Smithson [3]. The purpose of the present paper is to give a characterization of nested spaces analogous to Ward's characterization of trees [12]. Suppose (X, 6) is a space with a partial order < . We set L(x) = {y: y < x} and M(x) = {y: x < y}. A set A c X is an antiset in case no two elements of A are related. Further, the partial order is order dense in case x <y implies that there isaz E Xwithx <z <y. In this paper a continuum is a compact, connected T2-space and an arc is a continuum which contains exactly two nonseparating points. A tree is a continuum in which any two distinct points can be separated by the omission of a third point. Then a space X is nested in case it is arcwise connected, T2 and the union of every nest of arcs is contained in an arc. If X is a nested space and a, b E X, then there is exactly one arc in X with endpoints a, b. Using this fact we define a partial order < on X as follows. Let e E X be fixed. Then x < y iff x is an element of the arc with endpoints e, y. A number of properties of this partial order can be derived and
The structure of nested spaces
The structure of nested spaces is studied in this paper using such tools as branches, chains, partial orders, and rays in the context of semitrees. A classification scheme for various kinds of acyclic spaces is delineated in terms of semitrees. Several families of order compatible topologies for semitrees are investigated, and these families are grouped in a spectrum (inclusion chain) of topologies compatible with the semitree structure. The chain, interval, and tree topologies are scrutinized in some detail. Several topological characterizations of semitrees With certain order compatible topologies are also derived.
Effect of Calcium Glycerophosphate on Demineralization in an in vitro Biofilm Model
The aim was to investigate the anti-caries properties of calcium glycerophosphate (CaGP) using an in vitro bacterial flow cell model. Four flow cells, inoculated from a chemostat containing a seven-organism bacterial consortium, were pulsed with sucrose twice daily, to provide an acidic challenge and pH-cycling conditions. Blocks of enamel and dentine were mounted in each flow cell. In a study on the effect of CaGP concentration, CaGP was pulsed into three of the flow cells, at the same time as the sucrose, to give concentrations of 0.10, 0.25 and 0.50%. Water was pulsed into the fourth flow cell with the sucrose. Microradiography revealed a significant dose response of decreasing demineralization as CaGP concentration increased. Reductions at 0.25 and 0.5% were significant when compared to the control. A second study investigated the effect of timing of CaGP pulsing, relative to sucrose, on enamel and dentine demineralization. CaGP (flow cell concentration 0.2%), was pulsed 1 h before, during or 1 h after the sucrose pulse; a water control was employed. In enamel, pulsing CaGP before the sucrose reduced demineralization significantly compared to concurrent pulsing, which in turn gave a significant reduction compared to pulsing after sucrose, which did not reduce demineralization significantly compared to the water control. In dentine, CaGP reduced demineralization significantly only when pulsed before the sucrose. The findings suggest that in vivo, the anti-caries potential of CaGP may be greater if it is applied before a cariogenic challenge.