Im Kleinen Research Articles

Brown spaces and the Golomb topology

A Brown space is a topological space X such that for all non-empty open subsets U and V of X , we have cl cl . X X (U V ) ∩ ≠∅ ( ) It is clear that Brown spaces are connected and not completely Hausdorff. Given a b, , ∈ whose greatest common divisor is 1, we consider the arithmetic progression ( , ) = { : {0}}. P a b b an n G + ∈∪ The family G  of all such arithmetic progressions is a base for a topology Gτ on . In this paper we show that for every d ∈ , the set (1, ) GP d is a Brown space which is dense in ( , ). G  τ In particular, (, ) G  τ is a Brown space. We also show that for each prime number p and every natural number c, such that the greatest common divisor between p and c is 1, the set (,) GP pc is totally separated. We write some consequences of such result. For example that the space (, ) G  τ is not connected im kleinen at each of its points. This generalizes a result of Kirch AM.1 We also present a simpler proof of a result presented by Szczuka P.2 Some general properties of Brown spaces are also presented in this paper.

Topological properties in Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t≤1. In this paper, we study the following implication: if X has property P, then each Whitney block in C(X) has property P. We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m-mutual aposyndesis.

A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)

Abstract. In this paper, we investigate the relationships between the space X andthe hyperspace C ( X ) concerning admissibility and connectedness im kleinen. Thefollowing results are obtained: Let X be a Hausdorfi continuum, and let A 2 C ( X ).(1) If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ‰ IntK ‰ K ‰ U , then C ( X ) is connected im kleinen. at A . (2) If IntA 6= ; , then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ‰ IntK ‰ K ‰ U . (3) If X is connected im kleinen. at A , then A is admissible. (4) If A is admissible, thenfor any open subset U of C ( X ) containing A , there is an open subset V of X suchthat A ‰ V ‰ S U . (5) If for any open subset U of C ( X ) containing A , there is asubcontinuum K of X such that A 2 IntK ‰ K ‰ U and there is an open subset V of X such that A ‰ V ‰ S IntK , then A is admissible. 0. Introduction Let X be a Hausdorfi continuum, and let 2

ADMISSIBILITY AND CONNECTEDNESS IM KLEINEN IN HYPERSPACES

We investigate the relationships between the space X and the hyperspaces concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A, <TEX>$B{\in}C(X)$</TEX> with <TEX>$A{\subset}B$</TEX>. (1) If X is c.i.k. at A, then X is c.i.k. at B if and only if B is admissible. (2) If A is admissible and C(X) is c.i.k. at A, then for each open set U containing A there is a continuum K and a neighborhood V of A such that <TEX>$V{\subset}IntK{\subset}K{\subset}U$</TEX>. (3) If for each open subset U of X containing A, there is a continuum B in C(X) such that <TEX>$A{\subset}B{\subset}U$</TEX> and X is c.i.k. at B, then X is c.i.k. at A. (4) If X is not c.i.k. at a point x of X, then there is an open set U containing x and there is a sequence <TEX>$\{S_i\}^{\infty}_{i=1}$</TEX> of components of <TEX>$\bar{U}$</TEX> such that <TEX>$S_i{\longrightarrow}S$</TEX> where S is a nondegenerate continuum containing the point x and <TEX>$S_i{\cap}S={\emptyset}$</TEX> for each i = 1, 2, <TEX>${\cdots}$</TEX>.

ON ARCWISE CONNECTEDNESS IM KLEINEN IN HYPERSPACES

Let X be a space and 2X(C(X);K(X);CK(X)) denote the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, subcontinua) of X with the Vietoris topology. We investigate the relationships between the space X and its hyperspaces concerning the properties of connectedness im kleinen. We obtained the following: Let X be a locally compact Hausdor space. Let x 2 X. Then the following statements are equivalent: (1) X is connected im kleinen at x. (2) 2X is arcwise connected im kleinen at {x}. (3) K(X) is arcwise connected im kleinen at fxg. (4) CK(X) is arcwise connected im kleinen at fxg. (5) C(X) is arcwise connected im kleinen at {x}.

Homogeneous Suslinian Continua

AbstractA continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum X has the property that the set of points at which X is connected im kleinen is dense in X. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.

“Local–Global”: the first twenty years

This paper investigates how and when pairs of terms such as “local–global” and “im Kleinen–im Grossen” began to be used by mathematicians as explicit reflexive categories. A first phase of automatic search led to the delineation of the relevant corpus, and to the identification of the period from 1898 to 1918 as that of emergence. The emergence appears to have been, from the very start, both transdisciplinary (function theory, calculus of variations, differential geometry) and international, although the AMS-Gottingen connection played a specific part. First used as an expository and didactic tool (e.g. by Osgood), it soon played a crucial part in the creation of new mathematical concepts (e.g. in Hahn’s work), in the shaping of research agendas (e.g. Blaschke’s global differential geometry), and in Weyl’s axiomatic foundation of the manifold concept. We finally turn to France, where in the 1910s, in the wake of Poincare’s work, Hadamard began to promote a research agenda in terms of “passage du local au general.”

The Julia sets of basic uniCremer polynomials of arbitrary degree

Let P P be a polynomial of degree d d with a Cremer point p p and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets J P J_P . The red dwarf J P J_P are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing p p and the orbits of all critical images. The solar J P J_P are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and J P J_P is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of J P J_P and show that such sets J P J_P appear through polynomial-like maps for generic polynomials with Cremer points. Since known tools break down for d &gt; 2 d&gt;2 (if d &gt; 2 d&gt;2 , it is not known if there are small cycles near p p , while if d = 2 d=2 , this result is due to Yoccoz), we introduce wandering ray continua in J P J_P and provide a new application of Thurston laminations.

The solar Julia sets of basic quadratic Cremer polynomials

AbstractIn general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

The externally-fired gas-turbine (EFGT-Cycle) for decentralized use of biomass

The externally-fired gas turbine unites two advantages. On the one hand, the utilisation of the waste heat from the turbine in a recuperative process and, on the other, the possibility of burning “dirty” fuel. In particular, the EFGT opens a new option to utilise biomass for combined-heat-and-power and contributes to reduce greenhouse-gas emissions. A micro-gas turbine with 100 kW electric-output is chosen as an example to study the effects of temperature difference and pressure loss in the gas-to-air heat exchanger on cycle efficiency and power. The simulation calculations are performed with the code AspenPlus. In addition to cycle optimisation, the effect of low-calorific biogas on the combustion air ratio and the possibility of solar energy as a heat source for the EFGT are studied. For combusting biomass in an EFGT-Cycle, two alternatives are possible: First, a special, well-designed combustor for solid biomass, with a cyclone to reduce particles in the exhaust gas. Secondly, a gasifier with gas cleaning and a standard gas-burner. Waste heat from the process can be used for the gasification process, especially for drying and preheating the biomass. The detailed results are presented in the [Kautz M. Auslegung von extern gefeuerten Gasturbinen für dezentrale Energieanlagen im kleinen Leistungsbereich. Dissertation, Universität Rostock, Fakultät für Maschinenbau und Schiffstechnik; 4.11.2005.].

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace C n ( X ) , n < ∞ , is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of C n ( X ) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for C n ( X ) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C * -smoothness. Some new results are obtained for C ( X ) , the space of subcontinua of X. A class of continua X is given for which C n ( X ) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.

The Julia sets of quadratic Cremer polynomials

We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let f : U → U be a homeomorphism of a plane domain U and let T ⊂ U be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles θ , γ are K-equivalent if for some angles x 0 = θ , … , x n = γ the impressions of x i − 1 and x i are non-disjoint, 1 ⩽ i ⩽ n ; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense G δ -set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent.

Interiority of real functions

Conditions concerning openness of continuous real functions defined on topological spaces are studied. It is shown that for locally compact metrizable spaces interiority of the function at a point is equivalent to having no local extremum at this point if and only if the domain space is connected im kleinen at the point. Some related results are some obtained and open questions are posed. In this paper interrelations are studied between some conditions which concern openness of a real function (defined on a topological space), absence of any local extremum of this function at a given point, and local connectedness of the domain space. Additionally, although this investigation is related to the behaviour of real functions on metric or even on topological spaces in general, it has its roots in functional analysis. Namely, a theorem has been proved in [6] concerning sufficient conditions under which a Lipschitzian real function defined on a subset of a Banach space X has a Lipschitzian open extension over the whole space X . Subsequently, the third named author faced the problem of how to extend the above-mentioned result to (either all or some) topological or metric spaces X . A natural question that comes with this problem, and which seems to be interesting enough to warrant special attention, is what spaces admit an open (continuous) mapping onto the real line R? Even a surface observation of continuous functions from reals to reals suggests that openness, and lack of local extrema, may be linked. However, the existence of a local extremum, considered as an obstruction to openness, is a local property; therefore it violates openness at a point, or in a neighbourhood of the point. So, in these circumstances, a localization of openness should be considered rather than the global notion. Further study of the topic leads to the conclusion that local connectedness of the domain space plays a role in solving the problem. A careful analysis of a variety of considered conditions indicates that connectedness im kleinen at a point is a more adequate property than local connectedness at this point. The obtained results, and unsolved problems related to the topic, are presented below. Let us mention that interiority of real functions has been a subject of interest to topologists for more than fifty years (see, e.g., [9]). Let topological spaces X and Y be given and let f : X → Y be a function from X into Y (not necessarily continuous). The function f is said to be interior at a point p ∈ X (see [8, p. 149]) provided that for each neighbourhood U of p we have f ( p) ∈ int f (U). The function f is said to be open if it maps open subsets of the domain X onto open subsets of the range Y . Obviously,

Continuum theory. II.

1. APOSYNDETIC CONTINUA In this part of the paper, a continuum is a connected, metrizable, compact space. Through the whole paper a compactum is a compact Hausdorff space. The notion of aposyndetic continuum was introduced by F. B. Jones in 1941. A connected space X is called aposyndetic at a point x 6 X if, for each point y e X {z}, there exists a connected closed neighborhood of x, which does not contain y. Aposyndesis is a generalization of the notion of a connectivity im kleinen invented by R. L. Moore. A connected space X is called connected im klelnen at a point z if any neighborhood of z contains a connected closed neighborhood of x. Every continuum which is locally connected at a point x is connected im kleinen at this point, but the converse is not true. At the same time, a continuum which is connected im kleinen at each of its points is locally connected. On the other hand, aposyndesis is dual to the notion of a semi-local connectivity, introduced by Whyburn. A connected space X is said to be semi-locally connected (at a point x) if each of its points (the point x) has arbitrarily small neighborhoods whose complements consist of only a finite number of components. The duality of aposyndesis and semi-local connectivity can be illustrated by the following assertion. A connected compactum X is semi-locally connected at a point z if it ~s aposyndetic at each point y ~ x, and, conversely, X is aposyndetic at a point x if it is semi-locally connected at each point y ~ z. The invention of semi-locally connected continua was motivated by the following well-known Torhorst's theorem (1921): For every locally connected continuum in the plane, a boundary of each complementary domain (= a component of its complement) is locally connected. In 1939, Whyburn proved that, for every semi-locally connected continuum in the plane, a boundary

Absolute endpoints of chainable continua

An endpoint of chainable continuum is a point at which it is always possible to start chaining that continuum. Some endpoints appear to have the property that one is almost "forced" to start (or finish) the chaining at these points. This paper characterizes these "absolute endpoints", and this characterization is used to show that in a chainable continuum locally connected at p p is equivalent to connected im kleinen at p p .

β( X−{ x}) for X not locally connected

Let X be a metric continuum, xϵX, and Y = β( X −{ x}) − X. We prove that if Y is a continuum then Y is a decomposable continuum with one composant whenever X is not connected im kleinen at x. We give a sufficient condition for Y to be indecomposable. Finally we prove that Y is never a hereditarily indecomposable continuum although Y contains hereditarily indecomposable continua whenever X is hereditarily indecomposable.

Some functions on hyperspaces of continua

Two natural functions from the hyperspace 2X into the hyperspace C(C(X)) for an arbitrary metric continuum X are studied in this paper. Some characterizations of smoothness, local connectedness, connectedness im kleinen and of the property of Kelley of a continuum X are obtained in terms of continuity of these functions or their restrictions.

Cut points of 𝑋 and the hyperspace of subcontinua 𝐶(𝑋)

Let X X be a nondegenerate metric continuum and p 0 {p_0} a point with X = X 1 ∪ X 2 X = {X_1} \cup {X_2} , { p 0 } = X 1 ∩ X 2 \{ {p_0}\} = {X_1} \cap {X_2} , X 1 {X_1} and X 2 {X_2} continua. Denote by C ( X ) C(X) , C ( X 1 ) C({X_1}) and C ( X 2 ) C({X_2}) the hyperspaces of nonempty subcontinua of X X , X 1 {X_1} and X 2 {X_2} respectively. Theorem. C ( X ) C(X) is contractible if and only if C ( X 1 ) C({X_1}) and C ( X 2 ) C({X_2}) are contractible and either X 1 {X_1} or X 2 {X_2} is contractible im kleinen at p 0 {p_0} (a modification of connected im kleinen at p 0 {p_0} ). Theorem. Let X 1 {X_1} and X 2 {X_2} satisfy Kelley’s condition K K . Then C ( X ) C(X) is contractible when and only when either X 1 {X_1} or X 2 {X_2} is connected im kleinen at p 0 {p_0} . Examples are given.

More on connectedness im kleinen and local connectedness in 𝐶(𝑋)

Let X be a compact connected metric space and 2 X ( C ( X ) ) {2^X}(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. Let M ∈ C ( X ) M \in C(X) . If 2 X {2^X} is connected im kleinen at M, then C ( X ) C(X) is locally arcwise connected at M. A characterization of connectedness im kleinen in C ( X ) C(X) is given. Indecomposability of X is related to an absence of local connectedness in 2 X {2^X} and C ( X ) C(X) . An example is given of a continuum X and a subcontinuum M such that C ( X ) C(X) is connected im kleinen at M but not locally connected at M.

Connectedness im kleinen and local connectedness in 2XandC(X)

Let X be a compact connected metric space and 2X(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. In this paper the hyperspaces are investigated with respect to point-wise connectivity properties. Let MeC(X). Then 2X is locally connected (connected im kleinen) at M if and only if for each open set U containing M there is a connected open set V such that M <zV <zU (there is a component of U which contains M in its interior). This theorem is used to prove the following main result. Let Ae2 x. Then 2X is locally connected (connected im kleinen) at A if and only if 2 X is locally connected (connected im kleinen) at each component of A. Several related results about C(X) are also obtained. A continuum X will be a compact connected metric space. 2X(C(X)) denotes the hyperspace of closed subsets (subcontinua) of Xy each with the finite (Vietoris) topology, and since X is a continuum, each of 2X and C(X) is also a continuum (see [5]). One of the earliest results about hyperspaces of continua, due to Wojdyslawski [7], was that each of 2X and C(X) is locally connected if and only if X is locally connected. As a point-wise property, local connectedness is stronger than connectedness im kleinen, which in turn is stronger than aposyndesis. The author [1] has shown that if X is any continuum, then each of 2X and C(X) is aposyndetic. It is the purpose of this paper to investigate the internal structure of 2 X and C(X) with respect to these properties. In particular, we determine necessary and sufficient conditions (in terms of the neighborhood structure in X) that 2X be locally connected at a point and that 2X be connected im kleinen at a point. We also determine that C(X) has, in general, stronger point-wise connectivity properties that either 2X or X. For notational purposes, small letters will denote elements of X, capital letters will denote subsets of X and elements of 2X, and script letters will denote subsets of 2X. If A c X, then A* (int A) (bd A) will denote the closure (interior) (boundary) of A in X. Let x e X. Then X is locally connected (I.e.) at x if for each open set U containing x there is a connected open set V such that xe Vc U. X is connected im kleinen (c.i.k.) at x if for each open set U containing x there is a component of U which contains x in its interior. X is aposyndetic at x if for each y e X — x there is a