Peano Continuum Research Articles

Fixed point sets of 1-dimensional Peano continua

It is shown that every nonempty closed subset of a 1dimensional Peano continuum X is the fixed point set of some continuous self-mapping of X.

A Peano continuum homeomorphic to its own square but not to its countable infinite product

We give an example of a Peano continuum X with X 2 ≈ X {X^2} \approx X but X ∞ ≈ X X^\infty \approx X .

Fixed point sets of Peano continua

1* Introduction* A subset A of a topological space X is called a fixed point set of X if there is a (continuous) map /: X—>X such that f(x) = x iff x 6 A. If X is Hausdorff, then A is closed, and, clearly, every retract of X is a fixed point set of X. It is possible that a space X may have the property that each of its nonempty closed subsets is a fixed point set of X. The problem of determining which spaces have this property, called the complete invariance property by L. E. Ward, Jr. in [5], has been investigated by H. Bobbins, Helga Schirmer, and L. E. Ward, Jr. Some spaces known to have the complete invariance property include w-cells [1], dendrites [2], convex subsets of Banach spaces [5], compact manifolds without boundary [3], and all compact triangulable manifolds with or without boundary [4]. The general question as to what properties a space must satisfy to insure that it has the complete invariance property has not been resolved. In fact, in [5, p. 553] L. E. Ward, Jr. asks the following question. Does every Peano continuum have the complete invariance property*! The purpose of this note is to show that even acyclic Peano continua which possess higher order local connectedness need not have the complete invariance property. Indeed, for each positive integer n = 1, 2, , we give an example of an (n + l)-dimensional acyclic LC~ continuum Xn which fails to have the complete invariance property. Moreover, Xn contains an ^-dimensional sphere which is not a fixed point set of Xn.

Growth hyperspaces of Peano continua

For X a nondegenerate Peano continuum, let 2 X {2^X} be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that 2 X {2^X} is homeomorphic to the Hilbert cube. A nonempty closed subspace G \mathcal {G} of 2 X {2^X} is called a growth hyperspace provided it satisfies the following condition: if A ∈ G A \in \mathcal {G} , and B ∈ 2 X B \in {2^X} such that B ⊃ A B \supset A and each component of B meets A, then also B ∈ G B \in \mathcal {G} . The class of growth hyperspaces includes many previously considered subspaces of 2 X {2^X} . It is shown that if X contains no free arcs, and G \mathcal {G} is a nontrivial growth hyperspace, then G ∖ { X } \mathcal {G}\backslash \{ X\} is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace cc ( X ) {\text {cc}}(X) of closed convex subsets of a convex n-cell X.

A counterexample to a conjecture of A. H. Stone

A. H. Stone has offered a sequence, { S ( n ) ; n > 2 } \{ S(n);n > 2\} , of conjectures characterizing multicoherence for locally connected, connected, normal spaces. The conjecture S ( n ) S(n) is, “X is multicoherent if and only if X can be represented as the union of a circular chain of continua containing exactly n elements". It is known that S ( 3 ) S(3) always obtains and that S ( 6 ) S(6) obtains if the space is compact. In this paper, we construct a multicoherent plane Peano continuum C for which S ( 7 ) S(7) fails. Since S ( n + 1 ) S(n + 1) implies S ( n ) , n > 2 , S ( n ) S(n),n > 2,S(n) fails for C for all n > 6 n > 6 . Furthermore we show that for any integer n ⩾ 3 n \geqslant 3 there exists a plane Peano continuum for which S ( 2 n ) S(2n) obtains while S ( 2 n + 1 ) S(2n + 1) fails.

Semigroups with identity on Peano continua

Semigroups with identity on Peano continua

Semilattices on Peano Continua

A continuum is cell-cyclic if every cyclic element is an $n$-cell for some integer $n$. It is shown that every cell-cyclic Peano continuum admits a topological semilattice.

Semilattices on Peano continua

A continuum is cell-cyclic if every cyclic element is an n n -cell for some integer n n . It is shown that every cell-cyclic Peano continuum admits a topological semilattice.

Using brick partitionings to establish conditions which insure that a Peano continuum is a 2-cell, a 2-sphere or an annulus

Using brick partitionings to establish conditions which insure that a Peano continuum is a 2-cell, a 2-sphere or an annulus

Coarse uniformities on the rationals

There exist uniform spaces μ Q \mu Q homeomorphic with the space Q of rational numbers such that every homeomorphic uniform space admits a uniformly continuous homeomorphism upon μ Q \mu Q . If “homeomorphism” is replaced by “bijection", the resulting weaker property is equivalent to having as completion a Peano continuum. With “homeomorphism", a (countable) dense subspace of a Hilbert cube has the property, but not a dense subspace of an interval.

Separating certain plane-like spaces by Peano continua

Separating certain plane-like spaces by Peano continua

A fixed point theorem for plane continua

In this paper it is proved that every bounded arcwise connected plane continuum which does not separate the plane has the fixed point property. A set X is said to have the fixed point property if each continuous function ƒ on X into itself leaves some point fixed (that is, there is a point x belonging to X such that f(x)=x). The problem Must a bounded plane continuum which does not separate the plane have the fixed point has motivated a great deal of research in plane topology. K. Borsuk in 1932 proved that every Peano continuum which lies in the plane and does not separate the plane has the fixed point property [2]. Since that time, other general conditions have been found which insure that a plane continuum has this property. In 1967, H. Bell proved that every bounded plane continuum which does not separate the plane and has a hereditarily decomposable boundary has the fixed point property [ l ] . The following question is still outstanding. If a bounded plane continuum is arcwise connected and does not separate the plane, then must it have the fixed point property? Here an affirmative answer is given to this question by proving the following theorem. If M is a bounded arcwise connected plane continuum which does not have infinitely many complementary domains, then the boundary of M does not contain an indecomposable continuum. Throughout this paper 5 is the set of points of a simple closed surface (that is, a 2-sphere). The proof of the following theorem is based on techniques which are closely related to the folded complementary domain concept defined by F. Burton Jones [3, p. 173]. THEOREM 1. Suppose M is a continuum in Sy S—M does not have AMS 1970 subject classifications. Primary 54H25, 57A05, 47H10, 55C20; Secondary 54F22.

Can a 2-coherent Peano continuum separate 𝐸³?

The fact that there are unicoherent continua which separate E 2 {E^2} is well known, e.g., a circle with a spiral converging onto it is such a continuum. In this paper we extend this pathology by describing a Peano continuum which separates E 3 {E^3} and has the property that however it is written as the union of two unicoherent Peano continua, their intersection is unicoherent.

A partition theorem for collections of universal subcontinua

This paper is concerned with conditions under which a collection of universal subcontinua may be partitioned into finitely many subcollections, each of which has the finite intersection property. Throughout X will denote a Hausdorff topological space. A continuum A c X will be called a universal subcontinuum of (USC) X if A N B is a continuum for each continuum B c X. (Ordinarily it is assumed that X is a continuum, but this assumption is unnecessary for our purposes.) WALLACE, [3], studied USC under the title semi-chains. We are concerned with the following topological-combinatorial problem. If ~ is a collection of universal subcontinua, if n_->2 is a positive integer, and if of each n distinct elements of ct at least two have a non-empty intersection, does there exist an integer m <_- n -- 1 and m subcollections cq ..... at,, of ~, each of which has the finite intersection property, such that ct = ~ U -.- U ~t,,? We have not been able to answer this question in general, but in this paper we will answer it affirmatively for n = 3. The importance of the present problem lies in the fact that if ct is a collection of USC which has the finite intersection property then the intersection over ~ is a non-empty USC. The theorem presented here is of the type which has traditionally played a role in the study of the action of certain types of mappings on Hausdorff continua. It applies, for example, to collections of (i) compact nodal sets in a connected space, (ii) subcontinua of a hereditarily unicoherent continuum, and (iii) maximal cyclic elements in a peano continuum, [4]. It also applies to compact segments of the real line, where it overlaps a theorem of DEBRUNNER and HADWIGER [1] ; however, this special case can be treated by simpler methods than those used here. The following lemma is essentially lemma 1 of [2]. However, the proof is given here for the sake of completeness.

A note on means in Peano Continua

A note on means in Peano Continua

A note on means in Peano Continua

A note on means in Peano Continua

Topological transformation groups with a fixed end point

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

Connectivity retracts of finitely coherent Peano continua

Connectivity retracts of finitely coherent Peano continua

Fixed points in a transformation group

Proof of theorem, let /X, T, pi/ be transformation group, where X is Peano continuum with end point fixed under T

Stone's 2-Sphere Conjecture

space is separated by no point eliminates this trivial case. The property fails to hold for the 3-ball as can be seen by taking for the two domains the complements (in the 3-ball) of the right and left halves (respectively) of the equatorial disk. In fact, every cyclic Peano continuum in which property (U) holds true is a 2-sphere (Corollary 5).