On Peano spaces, with special reference to unicoherence and non-continuous functions

• Main Author: Hunt, John H. V
• Format: Dissertation
• Language: English
• Published: University of Warwick 1970
• Subjects: QA Mathematics

We have mentioned that each chapter in this thesis is conceived of as an independent paper, except for Chapter 3, which is a collection of results on non- continuous functions. Consequently each chapter contains a clearly marked introductory section, in which its back- ground and content are explained. In this abstract we shall summarize the remarks in these introductory sections. In chapter 1 we present an n-arc theorem for Peano spaces which is an extension of the theorem in §2 of [32], which Menger called the second n-arc theorem in [17]. Whereas in the second n-arc theorem n disjoint arcs are constructed joining two disjoint closed sets A and B, in chapter 1 we split the closed set A into n dis- joint closed subsets A1 A 2, ••• , An and give necessary and sufficient conditions for there to be n disjoint arcs joining A and B, one meeting each A1. At the end of chapter 1 we present a conjecture, which we have been able to verify in special cases. In [35] Whyburn proved a theorem concerning the weak connected separation of two non-degenerate connected closed sets A and B by a quasi-closed set L in a locally cohesive space X. In chapter 2 we show that A and B can in fact be taken as arbitrary closed sets in this theorem; that is, ,we remove the restriction of non-degeneracy and connectedness on A and B. In chapter 3 we study the circumstances under which a connectivity function is peripherally continuous. The study of the abstract relations between non- continuous functions was initiated by Stallings in [23]. In this paper he introduced the 1pc polyhedron and showed that a connectivity function was peripherally continuous on an 1pc polyhedron. Whyburn took up the study of non- continuous functions in [33]. [34] and [35]. He introduced the locally cohesive space, which is more general than the 1pc polyhedron, and proved that a connectivity function was peripherally continuous on a locally cohesive Peano space. For technical reasons, the locally cohesive space is not permitted to have local cut points. It is obvious, however, that on many Peano spaces having local, cut points a connectivity function remains peripherally continuous, In §2,3 of chapter 3 we formulate a sequence of properties Pn(X), which permit the space X to have local cut points, and we prove in each case that a connectivity function f : X →Y is peripherally continuous when X has property Pn(X). Each of these properties is an improvement on the last, and the final one, the U-space, satisfactorily incorporates the class of Peano spaces with local cut points on which we are able to prove that a connectivity function is peripherally continuous. An interesting feature of §3 of chapter 3 is provided by two "weak separation theorems," and more will be found about these in the introduction to chapter 3. In §4 of chapter 3 we show that a connectivity function is peripherally continuous on a locally compact ANR. This affirmatively answers a question that Stallings raised in [23]. The U-space that we have introduced in §3 of chapter 3 imposes a "unicoherence condition" in the space X (as do all the properties Pn(X) considered in §3, chapter 3). In §5 of chapter 3 we generalize the U-space to the S-space. This imposes a "multicoherence condition" on the space X, and we prove that a connectivity function is peripherally continuous on a cyclic S-space. We close chapter 3 by considering the question of placing weaker conditions than connectivity on the function f : X → Y which will still ensure that f is peripherally continuous. It is well known that if X is a unicoherent Peano continuum and A1, A 2, … is a sequence of disjoint closed subsets of X no one of which separates X, then Un=1 An does not separate X. In [28] van Est proved this theorem for the case where X is a Euclidean space of n dimensions. In chapter 4 we give an example which shows that this theorem does not hold if X is an arbitrary Peano space • In chapter 5 we provide a new angle to Lebesgue's covering lemma. We show that if the Lebesgue number ᵹ of an open covering U1, U2, •••• Un of a compact metric space X. ρ is finite. then it can be defined by the formula ᵹ = min ρ (E, F), where E and F are any compartments contained in no common U1 In chapter 6 we show that an involution on a cyclic Peano space leaves some simple closed curve setwise invariant. Whyburn has given a proof of R. L. Moore’s decomposition theorem for the 2-sphere in [31] (a refinement of this proof is presented in [36]). His proof is accomplished by showing that the decomposition space satisfies Zippin’s characterization theorem for the 2-sphere. In chapter 6 we present an alternative way of showing that the decomposition space satisfies Zippin's characterization theorem. Our proof closely follows Alexander's proof of the Jordan curve theorem as given by Newman in[21], and so consists of arguments that are well-known in another context. In [30] Whyburn gave a proof of the cyclic connectivity theorem. and in all subsequent appearances of this theorem in the literature Whyburn's proof has been used. Whyburn divided the proof of the theorem into three parts lemma 1, lemma 2, and the deduction of the theorem from lemmas 1 and2. In chapter 8 we give an alternative proof of lemma 1. Our proof is based on the fact that a cyclic Peano space has a base of regions whose closures do not separate the space, and it proceeds by an induction on a simple chain of these regions.