Chainable Continuum Research Articles

Embedding products of chainable continua

Bing [1 ] showed that every chainable continuum can be embedded in E2. A consequence of this is that if each of A1, * * *, A,. is a chainable continuum, then the topological product A1X * * * XA. can be embedded in E2n. This fact can also be derived from a theorem of Isbell [2]. This paper shows that the integer 2n can be replaced by n+ 1. The following example shows it cannot be replaced by n. EXAMPLE. For each integer n larger than 1 the product of n-I arcs and a sin(1/x) curve cannot be embedded in En. Such a continuum contains an n-cell and a subset disjoint from the n-cell with limit points in the interior of the n-cell. McCord [4] has proved an elegant embedding theorem which will be the principal tool used here. He defines a map f from a compact subset X of a metric space (E, d) to a compact subset Y of E to be approximable by homeomorphisms (relative to E) provided that for every e> 0 there is an open set U containing X and a 1 to 1 map ,g of U into E such that for all x in X, d(u(x), f(x)) <e.

Continua which have width zero

In a recent paper [7], the author showed that with each tree-like continuum there can be associated a non-negative number called the width of M, and it was shown that the plane, E2, does not contain uncountably many disjoint tree-like continua each having a positive width. This result is used here in establishing some conditions under which a tree-like continuum in E2 has width zero. There exists a treelike continuum, such as one which is the sum of a simple triod T and a ray spiralling around T, that has width zero but one of its subcontinua has a positive width. Some of the theorems presented here give conditions under which a tree-like continuum M has width zero hereditarily; that is, every subcontinuum of M has width zero.2 While such a continuum has a thinness property similar to that of a chainable continuum, there do exist in E2 tree-like continua, as indicated by Anderson [1], which have width zero hereditarily but are not chainable. The question in ?4 of [7] and Roberts' result [II] that every chainable continuum has uncountably many disjoint homeomorphic images in E2 suggest the following questions. If the tree-like continuum M is a subset of E2 and has width zero hereditarily, does there exist a sequence of disjoint continua in E2 converging homeomorphically to M? Does a tree-like continuum in E2 have uncountably many disjoint homeomorphic images in E2 if it has width zero hereditarily?3 These questions are not answered, but their converses are direct corollaries to some theorems in [7]. A tree-like continuum M in E2 has width zero hereditarily either if there exists a sequence of disjoint continua in E2 converging homeomorphically to M [7, Theorem 5] or if M has uncountably many disjoint homeomorphic images in E2 [7, Theorem 10]. In this paper, a compact connected metric space is called a continuum. Definitions of trees, chains, tree-like continua, and triods can be found in [6]. A definition of the width of a tree-like continuum is stated in [7], and the following property follows directly from this definition of width. A tree-like continuum M has width zero if, and

Each homogeneous nondegenerate chainable continuum is a coset space.

Let X be a compact Hausdorff homogeneous space and G be the full group of homeomorphisms from X to X. We denote the isotropy group at x by G.. If G is topologized by uniform convergence [1], then it is well known that G is a topological group, and the function 0: G/G-->X, defined by 0(gG,) =g(x) is one-one and continuous. It is an open problem whether 0 is a homeomorphism, i.e., whether X is a coset space. In this note we prove that each homogeneous nondegenerate chainable continuum M is such a space. Ford [2 ]2 has proved that every SLH (strongly locally homogeneous) completely regular Hausdorff space is a coset space. We will prove that M is not an SLH space but still a coset space. Bing [3] has proved that each homogeneous nondegenerate chainable space is a pseudo-arc. Therefore we only need to prove that the pseudo-arc is a coset space. In his papers [3; 4; 5], Bing gave a very beautiful treatment of the pseudo-arc. We will follow his definitions and notations, and we will not repeat all his definitions here. Description of the pseudo-arc M [4, p. 730]: Let P and Q be two points of a compact metric space and D1, D2, * * * a sequence of chains from P to Q such that for each positive integer i, (1) Di+, is crooked in Di, (2) no link of Di has a diameter of more than 1/2i and (3) the closure of each link of Di+, is a subset of a link of Di. Then M is the common part of D *, D2*, * * * , where D* denotes the sum of the links of Di. It is well known that the pseudo-arc is an indecomposable continuum [4]. DEFINITION. A space X is an SLH space if for any xEX, and neighborhood V of x, there is a neighborhood U of x, UC V, such that for any yG V, there is a homeomorphism g of X such that g(x) = y and g(z)=z if z EEU.

A Chainable Continuum no Two of whose Nondegenerate Subcontinua are Homeomorphic

A Chainable Continuum no Two of whose Nondegenerate Subcontinua are Homeomorphic

A chainable continuum no two of whose nondegenerate subcontinua are homeomorphic

A chainable continuum no two of whose nondegenerate subcontinua are homeomorphic

Each Homogeneous Nondegenerate Chainable Continuum is a Pseudo-Arc

Each Homogeneous Nondegenerate Chainable Continuum is a Pseudo-Arc

A fixed point theorem

Suppose that space is metric. A chain is a finite collection of open sets d1, d2, * * * , dn such that di intersects dj if and only if I i-jj I 1. If the elements of a chain are of diameter less than a positive number e, that chain is said to be an e-chain. A compact continuum is said to be chainable if for each positive number e, there is an e-chain covering it. R. H. Bing has called [1] such continua snake-like. In 1951 0. H. Hamilton showed [4] that every compact chainable continuum has the fixed point property; i.e., that if f is a continuous mapping of such a continuum M into itself, then some point of M is its own image underf. In the present paper it is shown that the Cartesian product of finitely many compact chainable continua has the fixed point property. Since arcs are compact chainable continua, this is a generalization of the Brouwer fixed point theorem. Two other examples of compact chainable continua are the closure of the graph of sin (1/x), 0 <x _1, and the pseudo-arc. Other examples are given in [1]. After reading the original manuscript of this paper, A. D. Wallace raised the question as to whether the Cartesian product of finitely many compact chainable continua is a quasi-complex (p. 323 of [6]). Rather surprisingly, the answer to this question is in the affirmative. A proof of this theorem is also given here. Since the Cartesian product of finitely many compact chainable continua is zero-cyclic, the fact that such products have the fixed point property is a special case of the Lefschetz fixed point theorem for zero-cyclic quasi-complexes. Since the author's original argument is of a very different nature, it is also given. Let En denote Euclidean n-space and Rn the set of all points of En whose distance from the origin is not greater than one. Let Sn-1 denote the set of all points of En whose distance from the origin is one. Let I denote the set of all points on the x-axis having abscissa x such that 0< x ?1, and let In denote the set of all points of En each of whose n coordinates xi satisfies 0 -xi_ 1. If P is a point of En having coordinates (x1, x2, * * *, xn) and t is a real number, by tP is meant the point of Es having coordinates (tx1, tx2, * . , txn).