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
A note on dendrites and trees
By a tree we mean a continuum ( = compact connected Hausdorff space) in which every pair of distinct points is separated by a third point. It is well known [4] that a dendrite is simply a metric tree. In this note we give a new characterization of dendrites and trees in terms of partially ordered spaces. Recall from [3] that a POTS ( = partially ordered topological space) is a topological space X endowed with a partial order, ^, which is semicontinuous in the sense that L(x) = [a'.a^x] and M(x)= [a'.x^a] are closed sets for each xGX. A partial order is order dense if x<y implies there is a z such that x<z<y. A chain of a partially ordered set is a subset which is linear with respect to the partial order. An element in a partially ordered set is minimal (maximal) if it has no proper predecessor (successor). Suppose that K is a locally connected continuum and e£PJ. Define a relation, ^, in K by x^y if, and only if, x = e, or x = y, or x separates e and y in K. From [3 ] we have
Use of laser scanning confocal microscopy for morphological taxonomy and the potential for digital type specimens (e-types)
The use of laser scanning confocal microscopy (LSCM) for creating taxonomic descriptions of copepods is investigated. A new technique is described, which employs a contour filter to process digital LSCM images, allowing taxonomic information to be quickly and accu- rately distilled into a simple illustration. LSCM allows the imaging of whole specimens, which can be rotated and viewed from any angle—a major benefit over light microscopy. Using this tech- nique, it is suggested that taxonomic descriptions can be rapidly produced in a fraction of the time required to produce similar descriptions using traditional light microscopy and hand drawing techniques. Good staining of specimens is, however, essential to produce accurate descriptions and more research is required in this area. The use of LSCM for morphological taxonomy shows great potential, not only for producing taxonomic descriptions, but also providing a comple - mentary adjunct to traditional type specimens in the form of 3D digital 'e-types' deposited in recognised international databases.