Another Proof that Any Compact Metric Space is the Continuous Image of the Cantor Set

1976 ◽  
Vol 83 (8) ◽  
pp. 646 ◽  
Author(s):  
Ira Rosenholtz
Keyword(s):  
Metric Space ◽  
Cantor Set ◽  
Continuous Image ◽  
Compact Metric Space

Perfect Images of Zero-Dimensional Separable Metric Spaces

1982 ◽  
Vol 25 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Jan Van Mill ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Cantor Set ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Surjection

AbstractLet Q denote the rationals, P the irrationals, C the Cantor set and L the space C − {p} (where p ∈ C). Let f : X → Y be a perfect continuous surjection. We show: (1) If X ∈ {Q, P, Q × P}, or if f is irreducible and X ∈ {C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X ∈ {P, C, L} and f is irreducible, then there is a dense subset S of Y such that f|f ← [S] is a homeomorphism onto S. However, if Z is any σ-compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q × C onto Z such that each fibre of the map is homeomorphic to C.

scholarly journals A theorem on homeomorphism groups and products of spaces

1969 ◽  
Vol 1 (1) ◽  
pp. 137-141 ◽  
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Cantor Set ◽  
Compact Metric Space ◽  
Compact Metric Spaces ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the Cantor set, C, onto itself. Let p: C → M be a map of C onto a compact metric space M, and let G(p, M) be is a group.The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn (xn) → y Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C) The present paper exhibits a structure theorem connecting these characterizing subgroups of H(C) and products of spaces: Let M1 and M2 be compact metric spaces. Then there are standard maps p: C → M1 × M2 and pi: C → Mi, i = 1, 2, such that G(p, M1 × M2) = G(p1, M1) ∩ G(p2, M2).

Weakly Confluent Mappings and Atriodic Suslinian Curves

1978 ◽  
Vol 30 (01) ◽  
pp. 32-44 ◽  
Author(s):  
H. Cook ◽  
A. Lelek
Keyword(s):  
Continuous Function ◽  
Branch Point ◽  
Metric Space ◽  
Related Result ◽  
Cantor Set ◽  
Topological Spaces ◽  
Compact Metric Space ◽  
Covering Theorem ◽  
The Given

There are theorems in which some classes of topological spaces are characterized by means of properties of mappings of these spaces into a single space. For example, it is well known that a compactum X is at most n-dimensional if and only if no mapping of X irto an (n + l)-cube has a stable value [5, Theorems VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if and only if no mapping of X into a figure eight is homotopically essential [1, Theorem 1, pp. 74-75; 8, p. 91]. By a curve we mean any at most 1-dimensional continuum; a continuum is a connected compactum; a compactum is a compact metric space, and a mapping is a continuous function. The aim of the present paper is to prove another theorem of this type. We distinguish a class of curves and show that it is characterized by imposing the condition that no weakly confluent mapping [13] can transform the given curve onto a simple triod (see 2.4). A related result is applied to a generalized branch-point covering theorem (see 3.2). In addition, two results are obtained in which we establish some characterizations of weakly confluent images and preimages of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are such images turn out to be identical with regular curves (see 1.3).

scholarly journals Almost minimal systems and periodicity in hyperspaces

2017 ◽  
Vol 38 (6) ◽  
pp. 2158-2179 ◽  
Author(s):  
LEOBARDO FERNÁNDEZ ◽  
CHRIS GOOD ◽  
MATE PULJIZ
Keyword(s):  
Metric Space ◽  
Necessary Conditions ◽  
Periodic Points ◽  
Cantor Set ◽  
Full Description ◽  
Compact Metric Space ◽  
Symmetric Products ◽  
Interval Maps ◽  
Admissible Sets ◽  
Induced Maps

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

Convergence of successive approximations to a stochastic fixed point

1980 ◽  
Vol 17 (1) ◽  
pp. 297-299
Author(s):  
Arun P. Sanghvi
Keyword(s):  
Fixed Point ◽  
Recent Result ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Compact Metric Space

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).

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