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.
Peripherally compact tree-like spaces are continuum-wise connected
The theorem stated in the title is proved, and examples are given to show the need for both peripheral compactness and tree-likeness. Two recent papers by Proizvolov ([2], [3]) provide much interesting and useful information about the structure of tree-like spaces. Among other facts, Proizvolov proves [2] that the density of a peripherally compact tree-like space equals its weight; each tree-like peripherally compact space X has a unique tree-like compactification which must have the same weight as that of X and [3] any two disjoint closed subsets of a peripherally compact tree-like space can be separated by a closed set; tree-like peripherally compact spaces are hereditarily normal and dyadic tree-like compact spaces are metrizable. Basic to the proofs of all these theorems is the theorem of the title. A paper by Gurin [1] is the reference cited, but it is not easy to see how to derive this assertion from his results. This note is devoted to an elementary proof of the assertion and some relevant examples. A topological space is said to be tree-like if it is connected, Hausdorff and every two points are separated by a third point. A peripherally compact space is a space with an open basis whose elements have compact boundaries. It is easy to see that a peripherally compact tree-like space has an open basis consisting of sets with finite boundaries. Therefore, each peripherally compact tree-like space has an open base consisting of connected sets with finite boundaries (see [4, p. 19]). PROOF OF THE THEOREM. Let X be a peripherally compact tree-like space and let p and q be two points of X. Define M to be the subset of X consisting of p, q and the points which separate p from q. Arguments given by Whyburn [4, pp. 42, 43 and 51] show that M is compact and Received by the editors September 6, 1971. AMS (MOS) subject class.fications (1970). Primary 54D05, 54D30, 54F15.