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
Recent Developments in Dendritic Spaces and Related Topics
Publisher Summary This chapter discusses the extension and unification of dendritic spaces. It discusses other classical characterizations of dendrites that lead to the definitions of dendrite-like spaces in the setting of connected Hausdorff spaces, and this gives rise to an extensive hierarchy whose inter-relationships are studied. The investigation is marked by its heavy reliance on the order-theoretic techniques suggested by the cutpoint partial order.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.