Continuous Surjection Research Articles

Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

Let f:X→Y be a continuous surjection of compact Hausdorff spaces. Byf⁎:M(X)→M(Y),μ↦μ∘f−1 and 2f:2X→2Y,A↦f[A] we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts:(1)If f⁎ is semi-open, then f is semi-open.(2)If f is semi-open densely open, then f⁎ is semi-open densely open.(3)f is open iff 2f is open.(4)f is semi-open iff 2f is semi-open.(5)f is irreducible iff 2f is irreducible.

The Mardešić conjecture for countably compact spaces

We shall show that if d and s are positive integers, Ki is a compact linearly ordered topological space for each i<d, X is a countably compact subspace of ∏i<dKi, Zj is an infinite Hausdorff space for each j<d+s, and f:X→∏j<d+sZj is a continuous surjection, then there exist at least s+1-many indexes j<d+s such that Zj is compact and metrizable. This improves the Mardešić Conjecture, which was proved by G. Martínez-Cervantes and G. Plebanek in [1].

Uncountable Family of 0-Rigid Continua that are Homeomorphic to Their Inverse Limits

It is a well-known fact that there are continua X such that the inverse limit of any inverse sequence {X,f_n} with surjective continuous bonding functions f_n is homeomorphic to X. The pseudoarc or any Cook continuum are examples of such continua. Recently, a large family of continua X was constructed in such a way that X is frac{1}{m}-rigid and the inverse limit of any inverse sequence {X,f_n} with surjective continuous bonding functions f_n is homeomorphic to X by Banič and Kac. In this paper, we construct an uncountable family of pairwise non-homeomorphic continua X such that X is 0-rigid and prove that for any sequence (f_n) of continuous surjections on X, the inverse limit varprojlim {X,f_n} is homeomorphic to X.

COUNTABLE SPACES WITH PEANO PROPERTY

In 1890, Giuseppe Peano published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square.In 1890, Giuseppe Peano published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square. According to the Hahn-Mazurkevich theorem the Hausdorff topological space $X$ is a continuous image of a unit segment $[0,1]$ if and only if when $X$ is compact, metrizable, connected, locally connected and nonempty. The Hausdorff continuous image of a segment is called {\it Peano space} or {\it Peano continuum}. Sierpinski proved that a connected compact metric space $X$ is a Peano continuum if and only if for every $\varepsilon&gt;0$ the space $X$ can be covered by connected sets of the diameter $\le\varepsilon$. Therefore, naturally arises question about the investigation of disconnected metric spaces $X$ for which there is a continuous surjection between $X$ and $X^2$. Sierpinski characterized rational numbers as a metric countable space without isolated points. Hausdorff described irrational numbers as a metric, separable, completely metrizable, zero-dimensional and nowhere locally compact space. It follows, in particular, that the square $\mathbb Q^2$ is a continuous image of the set $\mathbb Q$ and the square of irrational numbers is a continuous image of the set of irrational numbers. Thus, it would be interesting to find a description of other disconnected subsets of the real line, except those that are homeomorphic to $\mathbb Q$ or $\mathbb R\setminus Q$. In this article we will focus on countable sets such that the set of isolated points of which may not be empty. The main result is the following (see Theorem 2): the square of a countable regular topological space $X$ is its continuous image if and only if $X$ is not compact.

Metrizable subspaces of representation spaces

We show that various representation spaces for continua contain a homeomorphic copy of the harmonic sequence, so even though it is not a T0 space, it contains infinite metrizable subspaces. It remains an open question whether this representation space (topologized using continuous surjections) contains a closed subspace homeomorphic to the harmonic sequence. In fact, the representation space for continua topologized using various kinds of mappings (open, confluent, or monotone mappings, for instance) can be shown to have some closed points, but it remains open in many cases whether there are more than two closed points in most instances. Examples of closed points include the pseudo-arc and the universal pseudo-solenoid. Along the way, we find a few related results about metrizable subspaces of representation spaces for continua.

On 𝑇1- and 𝑇2-productable compact spaces

Abstract We prove that if there exists a continuous surjection from a metric compact space 𝑋 onto a product X × T X\times T , where 𝑇 is a T 1 T_{1} second countable topological space which has the cardinality of the continuum, then there exists a surjection from 𝑋 onto the product X × [ 0 , 1 ] X\times[0,1] , where the interval [ 0 , 1 ] [0,1] is equipped with the usual Euclidean topology.

A geometric Vietoris-Begle theorem, with an application to convex subsets of topological vector lattices

We show that if L is a topological vector lattice, u:L→L is the function u(x)=x∨0, C⊂L is convex, and D=u(C) is metrizable, then D is an ANR and u|C:C→D is a homotopy equivalence, so D is contractible and thus an AR. This is proved by verifying the hypotheses of a second result: if X is a connected space that is homotopy equivalent to an ANR, Y is an ANR, and f:X→Y is a continuous surjection such that, for each y∈Y and each neighborhood V⊂Y of y, there is a neighborhood V′⊂V of y such that f−1(V′) can be contracted in f−1(V), then f is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.

A generalization of Whyburn's theorem, and aperiodicity for Abelian C⁎-inclusions

Let j:Y→X be a continuous surjection of compact metric spaces. Whyburn proved that j is irreducible, meaning that j(F)⊊X for any proper closed subset F⊊Y, if and only if j is almost one-to-one, in the sense that{y∈Y:card(j−1(j(y)))=1}‾=Y. In this note we prove the following generalization: There exists a unique minimal closed set K⊆Y such that j(K)=X if and only if{x∈X:card(j−1(x))=1}‾=X. Translated to the language of operator algebras, this says that if A⊆B is a unital inclusion of separable abelian C⁎-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian C⁎-algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwaśniewski-Meyer).

On linear continuous operators between distinguished spaces $$C_p(X)$$

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space $$C_{p}(X)$$ is distinguished if and only if X is a $$\Delta $$ -space. If there exists a linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ and $$C_p(X)$$ is distinguished, then $$C_p(Y)$$ also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator $$T:C_p(X) \rightarrow C_p(Y)$$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $$C_p([1,\alpha ])$$ , where $$\alpha $$ is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping $$T:C_p([1,\alpha ]) \rightarrow C_p(Y)$$ is given. We also observe that for every countable ordinal $$\alpha $$ all closed linear subspaces of $$C_p([1,\alpha ])$$ are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of $$\Delta $$ -spaces we prove that a linear continuous surjection $$T:C_p(X) \rightarrow C_k(X)_w$$ , where $$C_k(X)_w$$ denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact $$X \subset {\mathbb {R}}^n$$ ).

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff spaceXXis aΔ\Delta-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex spaceCp(X)C_{p}(X)is distinguished. Continuing this research, we investigate whether the classΔ\DeltaofΔ\Delta-spaces is invariant under the basic topological operations.We prove that ifX∈ΔX \in \Deltaandφ:X→Y\varphi :X \to Yis a continuous surjection such thatφ(F)\varphi (F)is anFσF_{\sigma }-set inYYfor every closed setF⊂XF \subset X, then alsoY∈ΔY\in \Delta. As a consequence, ifXXis a countable union of closed subspacesXiX_isuch that eachXi∈ΔX_i\in \Delta, then alsoX∈ΔX\in \Delta. In particular,σ\sigma-product of any family of scattered Eberlein compact spaces is aΔ\Delta-space and the product of aΔ\Delta-space with a countable space is aΔ\Delta-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].LetT:Cp(X)⟶Cp(Y)T:C_p(X) \longrightarrow C_p(Y)be a continuous linear surjection. We observe thatTTadmits an extension to a linear continuous operatorT^\widehat {T}fromRX\mathbb {R}^XontoRY\mathbb {R}^Yand deduce thatYYis aΔ\Delta-space wheneverXXis. Similarly, assuming thatXXandYYare metrizable spaces, we show thatYYis aQQ-set wheneverXXis.Making use of obtained results, we provide a very short proof for the claim that every compactΔ\Delta-space has countable tightness. As a consequence, under Proper Forcing Axiom every compactΔ\Delta-space is sequential.In the article we pose a dozen open questions.

A fiberable continuum which is not nontrivially productable II

We construct a metric continuum X such that there exists a continuous surjection F from X onto X with the inverse image of every point of size of the continuum and such that for no nontrivial topological space T there exists a surjection from X onto X×T. In particular, F cannot be a composition of a surjection from X onto X×T and the projection onto X, where T is a topological space of size of the continuum with at least one nonempty open set not equal to the whole space. This is a sharpening of the result obtained in [2].

A fiberable continuum which is not nontrivially productable

We construct a continuum X⊆R3 such that there exists a continuous surjection f from X onto X with the inverse image of every point being of the same size as the continuum and such that for no topological space T whose every point has a nontrivial neighbourhood there exists a surjection from X onto X×T. In particular, f cannot be a composition of a surjection from X onto X×T and the projection onto X, where T is of the same size as the continuum and satisfies the above neighbourhood condition.

Martin boundary covers Floyd boundary

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

Periodic solutions of the Poincaré functional equation: Existence

Let I be a compact real interval, k>1 an integer and f:I→I a continuous surjection such that: (i) f achieves its absolute extrema at exactly k+1 points, and on the k remaining intervals f is strictly monotone, (ii) the set of all points among which the iterations of f attain their absolute extrema is dense in I. We show that the above conditions on f are necessary and sufficient for the Poincaré functional equation ψ(kx)=f(ψ(x)) to have a cosine-like solution ψ:R→R.

Sensitivity for topologically double ergodic dynamical systems

&lt;abstract&gt;&lt;p&gt;As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.&lt;/p&gt;&lt;/abstract&gt;

Spaces of algebraic measure trees and triangulations of the circle

In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch point map which assigns to each triple of points their branch point. We give an axiomatic definition of algebraic trees, define a natural topology, and equip them with a probability measure on the Borel-$\sigma$-field.Under an order-separability condition, algebraic (measure) trees can be considered as tree structure equivalence classes of metric (measure) trees (i.e.\ subtrees of R-trees). Using Gromov-weak convergence (i.e.\ sample distance convergence) of the particular representatives given by the metric arising from the distribution of branch points, we define a metrizable topology on the space of equivalence classes of algebraic measure trees. In many applications, binary trees are of particular interest. We introduce on that subspace with the sample shape and the sample subtree mass convergence two additional, natural topologies. Relying on the connection to triangulations of the circle, we show that all three topologies are actually the same, and the space of binary algebraic measure trees is compact. To this end, we provide a formal definition of triangulations of the circle, and show that the coding map which sends a triangulation to an algebraic measure tree is a continuous surjection onto the subspace of binary algebraic non-atomic measure trees.

On genericity of shadowing in one dimension

We show that shadowing is a generic property among continuous maps and surjections on a large class of locally connected one-dimensional continua.

A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

AbstractThe Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

The isomorphism class of the shift map

The shift map, denoted σ, is the self-homeomorphism of ▪ induced by the successor function n↦n+1 on ω. We prove that the isomorphism classes of σ and σ−1 cannot be separated by a Borel set in H(ω⁎), the space of all self-homeomorphisms of ω⁎ equipped with the compact-open topology.Van Douwen proved it is consistent for σ and σ−1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ−1 may not be isomorphic, there is no simple topological property that distinguishes them.As a relatively straightforward consequence of the main theorem, we deduce that OCA+MA implies the set of continuous images of σ fails to be Borel in H(ω⁎). (Here a “continuous image” of σ is meant in the sense of topological dynamics: any h∈H(ω⁎) such that q∘σ=h∘q for some continuous surjection q:ω⁎→ω⁎.) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of H(ω⁎).

Fuzzy Fibrewise Homotopy

In this disquisition, the notion structures is introduced. The intend of this article is to study the notions of homotopy, path homotopy and fundamental group via structure which is a triplet consisting of two fuzzy topological spaces and a continuous surjection between them. Many properties concerning these concepts are provided.