Refinable Maps Research Articles

Uniformly refinable maps

We introduce the notion of uniformly refinable map for compact, Hausdorff spaces, as a generalization of refinable maps originallydefined for metric continua by Jo Ford (Heath) and Jack W. Rogers, Jr., Refinable maps, Colloq. Math., 39 (1978), 263-269.

Proximately chain refinable functions

We define proximately chain refinable functions as a generalization of refinable maps and investigate some of their properties for Hausdorff paracompact spaces. We prove that the proximate fixed point property is preserved by proximate near homeomorphisms in paracompact Hausdorff spaces. This generalizes a previous result of E. Grace.

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G. We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.

Extension Dimension and Refinable Maps

Extension dimension is characterized in terms of ω-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some infinite-dimensional properties.

Spaces of discrete shape and c -refinable maps that induce shape equivalences

Spaces of discrete shape and c -refinable maps that induce shape equivalences

Generalized refinable maps and images of FANR's

In this paper we study certain shape and homotopy properties of refinable and weakly refinable maps. In particular we give a sufficient condition for a weakly refinable map to be a homotopy domination when the codomain is an ANR. We also characterise those weakly refinable maps which preserve FANR's and using a result of Boxer we characterise the weakly refinable maps which preserve ANR's.

Stability of almost commutative inverse systems of compacta

Recently, S. Mardešić and L.R. Rubin have defined approximate inverse systems of compacta. In these systems the bonding maps commute only up to certain controlled values. In this paper it is shown that such systems are stable in the sense that small perturbations of bonding maps yield again an approximate system and do not affect the limit space. In particular, an inverse system of near homeomorphisms can be replaced by an approximate inverse system of homeomorphisms having the same limit space. In such a system projections from the limit space need not be (near) homeomorphisms, but are always refinable maps.

Refinable maps and 𝜃_{𝑛}-continua

Many properties of continua, e.g., irreducibility, span zero, and various kinds of aposyndesis and unicoherence, are preserved by refinable maps or their inverses. The purpose of this paper is to consider the extent to which the property of being a θ n ′ {\theta ’_n} -continuum is preserved by refinable maps or by inverses of refinable maps. One consequence of this study is to generalize results of the first author where the domain or range is a graph. Several of his results are corollaries of ours.

Stability of the fixed point property and universal maps

We give a stability condition for the fixed point property in terms of the fundamental metric and the metric of continuity introduced by K. Borsuk. This condition is equivalent to that originally given by V. Klee but reflects richer properties. We introduce the notion of a proximately universal map and study many of its properties. Relationships among proximately universal maps and some generalized refinable maps introduced by E. E. Grace are studied. In particular we prove that every weakly refinable map $r:X \to Y$ is proximately universal whenever $X$ has the proximate fixed point property. This generalizes a result of Grace.

Stability of the Fixed Point Property and Universal Maps

In this interesting paper, the author gives a stability condition for the fixed point property in terms of K. Borsuk's fundamental metric on a hyperspace of a compact metric space. This condition is equivalent to that originally given by V. L. Klee [Colloq. Math. 8 (1961), 43–46] but it reflects richer properties. By replacing exact conditions with their proximate analogues, the author introduces a notion of proximately universal maps and studies many of their properties. In particular, he investigates their preservation under composition with weakly refinable and refinable maps to get improvements of results of E. E. Grace [Proc. Amer. Math. Soc. 98 (1986), no. 2, 329–335] and C. W. Ho [Fund. Math. 111 (1981), no. 2, 169–177].

Property C, refinable maps and dimension raising maps

We show that refinable maps defined on compacta preserve Property C. H. Kato has proved the analogous result for weakly infinite dimensional spaces. We also show that if f f is a map from a compact C C space X X onto a non C C space Y Y , then the set of points in Y Y with an uncountable number of preimages is a space that does not have Property C.

A note on span under refinable maps

All spaces considered In this note are metric, and all maps are continuous functions. A compactum is a compact metric space. A continuum is a connected compactum. In [1], Ford and Rogers defined a mapr: X-*Y from a compactum X onto a compactum Y to be refinableif for each £>0, there is an s-map/: X-+Y from X onto Y whose distance from r is less than s. Refinable maps are useful in continuum theory, and many properties in continuum theory are preserved by refinable maps. For example, decomposability [1], aposyndesis [2], property [k], irreducibility,hereditary indecomposability and being the pseudo-arc [6] (see for other properties [4] and [5]). Lelek [8] defined the surjective span of a continuum X, a*(X), (resp. the surjective semi-span, o*(X)) to be the least upper bound of all real numbers a with the following property; there exist a continuum C and maps ft,ft: C-*X such that /1(C)=X=/8(C)(resp. /a(C)=J^) and dist(/i(c),/2(c))^a for every cg C. The span a(X) and the semi-span <?0{X) of X are defined by the formulas;

Refinable maps in dimension theory

This paper studies properties of refinable maps and contains applications to dimension theory. It is proved that refinable maps between compact Hausdorff spaces preserve covering dimension exactly and do not raise small cohomological dimension with any coefficient group. The notion of a c-refinable map is introduced and is shown to play a comparable role in the setting of normal spaces. For example, c-refinable maps between normal spaces are shown to preserve covering dimension and S-weak infinite-dimensionality. These facts do not hold for refinable maps.

A Remark on Refinable Maps and Calmness

It is shown that if $r:X \to Y$ is a refinable map between compacta and $Y$ is calm, then $r$ is a shape equivalence. As a corollary, if $r:X \to Y$ is a refinable map between compacta and either $X$ or $Y$ is ${S^n}$-like $(n \geqslant 1)$, then $r$ is a shape equivalence, where ${S^n}$ denotes the $n$-sphere.

A remark on refinable maps and calmness

It is shown that if r : X → Y r:X \to Y is a refinable map between compacta and Y Y is calm, then r r is a shape equivalence. As a corollary, if r : X → Y r:X \to Y is a refinable map between compacta and either X X or Y Y is S n {S^n} -like ( n ⩾ 1 ) (n \geqslant 1) , then r r is a shape equivalence, where S n {S^n} denotes the n n -sphere.

Aposyndesis and coherence of continua under refinable maps

Aposyndesis and coherence of continua under refinable maps

A note on infinite-dimension under refinable maps

It is shown that refinable maps preserve weak infinite-dimension, but not strong infinite-dimension.

Refinable maps and generalized absolute neighborhood retracts

This paper studies some properties which are preserved by refinable maps when the domain is a certain kind of generalized ANR called a quasi-ANR. Among those properties are the properties of being a quasi-ANR and the disjoint n-cell property. An alternative characterization of quasi-ANR's as surjective approximative neighborhood extensors is also obtained in Theorems 3 and 4. These characterizations are analogs of similar characterizations in ANR theory.

A note on refinable maps and quasi-homeomorphic compacta

A note on refinable maps and quasi-homeomorphic compacta

Refinable maps in the theory of shape

Refinable maps in the theory of shape