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
Mobs and semitrees
In (1957) Ward characterized trees as being certain compact idempotent commutative monotone mobs. The purpose of the present note is to obtain a similar characterization for the semitrees as studied by Muenzenberger and Smithson (1973, 1975).
A QUASI ORDER CHARACTERIZATION OF SMOOTH CONTINUA
A quasi order on a topological space X is a reflexive and transitive binary relation rg. If this relation is also antisymmetric it is called a partial order. The quasi order ^ is closed if {(x, y)e X x XI x ^ y} is a closed subset of the product space X x X. For each xe X, the set L(x) = {y e X | y ^ x) (respectively, M{x) = {y 6 XI x ^ y}) is called the set of predecessors (respectively, successors) of x. Let E(x) = L(x) Π M(x) and note that ^ is a partial order if and only if each E{x) is a singleton. In case ^ is closed, the sets L(x), M(x), and E(x) are closed subsets of X. If x ^ y and x £ E(y) we write x < y. The quasi order ^ is order dense if whenever x < y, there exists ze X such that x < z < y. Let S be a subset of X. An element z e S is a zero of S if 2 ^ # for each xeS. If x ^ y or y ^ x for all x, y e S, then S is called a