LOCAL CONNECTIVITY AND MAPS ONTO NON-METRIZABLE ARCS

Three classes of locally connected continua which admit sufficiently many maps onto non-metric arcs are investigated. It is proved that all continua in those classes are continuous images of arcs and, therefore, have other quite nice properties.


INTRODUCTION
Let C denote the class of all Hausdorif continuous images of ordered continua.In the last three decades the class C has been studied extensively by a number of authors (see e.g.[2], [4], [6][7][8], [11][12][13], [16][17][18][19][20][21][22], [26] and [27]).Two results from this study have suggested that the investigation could naturally be extended to the larger class TiM of all rim-metrizable, locally connected continua.Namely, (1) in [8] in 1967 Mardeid proved that each element of C has a basis of open Fa-sets with metrizable boundaries, and (2) in [4] in 1991 Grispolakis, Nikiel, Simone and Tymchatyn showed that if a set P is irreducible with respect to the property of being a compact set which separates the element X of C, then P is metrizable.I his 1989 thesis [23] and two subsequent papers [24] and [25] Tuncali began an investigation of the class 7M and continuous images of elements of that class.He showed that Treybig's product theorem of [18] which holds in C is no longer valid in RM.However, he proved that Mardeid's theorem for C on preservation of weight by light mappings is true in RM, [25].He also considered the class T.s of all rim-scattered, locally connected continua, and the class Rc of all tim-countable, locally connected continua.Later, Nikiel, Tuncali and Tymchatyn gave an example to show that R.c is not a subclass of t?, [15].Then, recently the authors of this paper showed the the continuous image of an element of 7ZM need not be in R.M, [14].Furthermore, Drozdovsky and Filippov proved in [3] that Rs is a larger class of spaces than/Zc.Also, in 1973 Heath, Lutzer and Zenor, [5], showed that every linearly ordered ordered topologic space and each of its Hausdorff continuous and closed images are monotonically normal.In [10] in 1986 Nikiel asked if every monotonically normal compactum is the continuous image of a compact ordered space.That problem still remains open.In what follows we let R.MN denote the class of monotonically normal, locally connected continua.Our first result is th,e following: THEOREM 1.If X E M J S U .MN and for each pair of points a, b E X there exists a continuous onto map f: X [c, d] such that f(a) c, f(b) d and [c, d] is a non-metrizable arc, then X ?.
We note that a large class of examples satisfying the properties of X above can be con- structed as follows: In [1] in 1945 Arens studied the class of linear homogeneous continua, that is the class of arcs which are order isomorphic to each of their subarcs.Arens showed, that up to a homeomorphism, there exist at least R1 members of , including the real numbers interv [0,1].Thus, some spaces X as in Theorem 1 could be obtained by pasting together copies of any Z .
If a subset B of a space P contains no dense-in-itself, non-empty subset, we say that B is scattered.
In this paper the definition of monotone normality we use is an equivalent one given in Lemma 2.2 (a) of [5].It says that a space P is monotonically normal provided there is an operator G which assigns to ear ordered pair (S, T) of mutually separated subsets of P an open set G(S, T) such that (i) S C G(S,T) C cl(G(S,T)) C P-T, and (ii) if (S',T') is also a pair of mutually separated sets such that S C S' and T' C T, then G(S, T) c (S', T').
For each positive integer n let H, be a component of G, R which intersects bd (Rl)   and bd(U), and let s. H. Iq bd(R1) and t, H, N bd(U).Let H0 denote the limiting set of the sequence H, H2, H3,...; which by definition is the set of all x such that every open set containing x intersects infinitely many sets H,.
We shall show that some component of H0 contains s, }.If not, then H0 is the union of two mutually separated sets S and T such that s S and T.
There exist disjoint open sets V and W so that S C V and T C W. Then (sn, t,) belongs to V x W for infinitely many n.
Our proof now divides into three cases.CASE 1. X 7M.For each n > 2 let M, denote a metrizable closed set lying in X-Urn--1 Ht such that if 1 < < j < n, then H, and Hj are selarated in X by M,.Let D. denote a countable set dense in M, for n 2, 3,...We intend to show that f 0=2 Dr) is dense in [c, d], which would mean that [c, d] is separable, and therefore metric, a contradiction.Let x (5 ]c, d[ and let c < u < x < v < d in the natural ordering of [c, d].The components of f-(]u, v D which have limit points in both f-(u) and f-(v) can be labeled P,P2,... ,P,o.
Let No be an integer such that if >_ No then s,, f-l([c,u[) and t,, f-(]v,d]).There exist two of No,No + 1,... ,No + no, say and j, such that Ha. and Ha, both intersect the same Pt, which must then intersect some D,.. Therefore, 0=2 f(D) intersects ]u, v[. which intersects f-(c') and f-(d'), and let Q0 denote the limiting set of Q, Q2, Q3,... We note that some component of Q0 intersects both f-(c') and f-l(d') since every map onto an arc is weakly confluent.
We have to consider some subcases.CASE 3A.[c', d'] contains uncountably many mutually exclusive open sets.CASE 3A1.[c',d'] does not satisfy the first axiom of countability.Thus, without loss of generality, assume that there is a subset {da a < w of [c', d'] such that a < a2 implies that da, < da2 in [c', d'], and da d'.
Let K0 denote the limiting set of K1, K2, K3,... Let Q denote a component of K0 which intersects both f-(c') and f-(d').For each a < Wl let Wa denote a connected open set such that Wa contains a point Xa of Qcl f-(]da, da+[), and Wa c f-(]da, do,+[).
There exists a positive integer no and a cofinal subsequence {dan } of da such that Kn0 # @ for all a0.For each 7 < wx let L, denote the closure of the set 0_>, Wa.Let L -<,01 Lv.Observe that if y 6-L, then each open neighborhood of y intersects uncountably many sets War.Let W be a component of L. Note that WClK, # 0 # QclW and W C f-(d').
Thus, W is a non-degenerate continuum.
Let {Mo, M }.Now suppose that , has been chosen and consists of 2 mutually exclusive connected open sets such that if G,G' q , and G # G', then C G ''7 1 and G Cl I # l # G' N W. For each G' 6-, let G and G be mutually exclusive connected open sets such that G o Cl G1 GUG C G' and GW # 0 # G W. Let ,+ {F-F G or F G for some G' 6-g,}.For each n let H: n and let H [n=l gn" There exists $0 < w such that G' Cl f-(do for each G' 6-U= .There exists a closed scattered set c in X which separates f-([c, d0] from f-(d').However, CglH contains a perfect set because c Cl H can be mapped onto a Cantor set, and it is well known that a scattered set cannot be mapped continuously onto a perfect set.This is a contradiction.CASE 3A2. [c', d'] satisfies the first axiom of countability at each point.Let {]ca, a < w } denote an uncountable collection of mutually exclusive open intervals in ]c', d'[.Using the local connectivity of X we find that for each a there exists only a finite number, say ha, of components of f-(]ca, da[) which have limit points in both f-(ca) and f-l(d,,).Some integer No n= repeats for uncountably many a,'s; so we may suppose without loss of generality that n,, No for each a < w.
There exists a closed scattered set S such that S separates K, from Kj for each pair i, such that 1 _< < j _< No + 1.Thus, since for each a, each set K, where 1 _< _< No + has the property that some component of K, N f-l(]ca, d,[) has limit points in both f-'(ca) and f-l(d,), it follows that S must intersect each f-1 (]ca, da[).
For each positive integer n let L', [.J,,, and let L' --1L.We find that S f-(L') contains a perfect set, a contradiction.CASE 3B.[c,d] is not metrizable and does not contain uncountably many mutually exclusive open sets (i.e., it is a Souslin line).Thus, [c', d'] satisfies the first axiom of countability.
If there exists a collection of metrizable open intervals whose union is dense in [c , d], we find that [c', d] is metrizable since it is separable.Hence, without loss of generality we may assume that [c , d'] contains no metrizable subinterval.
Similarly as above, for each Ix, y[ C [d, d'] we let nz denote the number of components of f-(]z,y[) with limit points in both f-(x) and f-(y).CASE 3B.Suppose there exists a positive integer No and a subinterval Ix, y[ of such that ifx _< z < w _< y, then n, _< No. Let S be aclosed scattered set such that if 1 _< < j _< No + 1, then S separates K, from Kj.Using the ideas from Case 3A we find that if x _< z < w _< y, then S ]'-(]z, w[) # .Therefore, f(,.S) :3 Ix, y], which contradicts the well-known fact that a scattered compactum can not be mapped onto a perfect set.
CASE 3B2.Assume that for every Ix, y[ C [c', d'] there exists an interval ]z, w[ C Ix, y[ such that n,o > n=.
For each positive integer n let ,, be maximal relative to the property of being a collection of mutually exclusive open intervals lying in [d, d'] such that if Ix, y[ ,, then nffi n.Note that each ,, is at most countable.Let S,, denote the set of all end-points of intervals which belong to ,,.We are going to show that U,,__ Sn is dense in [c', d'], and thus obtain a contradiction.

Let Ix, y[ C [d, d].
There exists ]z, w[ C Ix, y[ such that n > nz.Thus, x # z or y # w.
The consideration of subcases 1, 2 and 3 is concluded and we return now to the main proof.
Since X is hereditarily locally connected, it is the continuous image of an arc by [12].THEOREM 2. If X is as in Theorem 1, then (a) X is rim-finite, (b) every subcontinuum G of X has the property that some point or a pair of points separates G, and (c) each closed set irreducible with respect to the property of being a compact set which sepa- rates X is metrizable.
Given a locally connected continuum X, for each pair of distinct points a, b of X let IX, a, b] denote the class of all continuous maps f X P such that P f(X) is a non-metric arc with end-points c and d and f(a) c and f(b) d.Also, introduce a relation on X in the following way: a b if and only if a b or [X, a, b] .THEOREM 3. Suppose that X is a locally connected continuum.Then is an equiv- alence relation on X, and if X also satisfies the first axiom of countability, then equivalence classes of are closed and the set ' of equivalence classes of is upper semi-continuous.
PROOF. is easily seen to be reflexive and symmetric, so suppose that a b and b c hold, but that there exists f E IX, a, c] such that f(X) is a non-metric arc [d, e]     Let us now show that each equivalence class G and suppose that x E -G.There exists a countable basis U1, U,... of open neighborhoods of x in X and a sequence x,z2,.., of points of G such that x, U, for 1,2,... Let f" X --, [c,d] be a continuous map onto a non-metric arc [c,d], where f(zl) c and f(z) d.
Since each [y(),y(z)] is a metric subarc of [c,d], it follows that [c,d] is the closure of a countable union of mettle ares.Consequently, [c,d] is separable, and therefore mettlzable, a contradiction.Thus G is closed in X.It remains to show that is upper semi-continuous if X is first countable.Let the element G of ' be a subset of an open set U. Suppose that for each open set V such that G C V C U, there is an element Gv of so that V Iq Gv l and Gv .U .Thus, for some point x of G there is a countable basis U, U,... of open neighborhoods of z such that for each U, there is an element G, of " with the property that G, fq U, } Gi tq (X U).
There is a point y of X U so that every neighborhood of y intersects Gi for infinitely many i.We may assume without loss of generality that there exists y, G, [q (X U) for each i, and that the points yi converge to y.Let z, E Ui f'l G, for 1, 2,...There exists f [X,z, y] such that f X [c,d], where [c, d] is a non-metric arc, f(x) c and f(y) d.Since the points f(z,) converge to c, and the points f(yi) converge to d, and each arc If(z,), f(y,)] is metric, we find that [c,d] is metric a contradiction.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

CASE 1 .
f(b) d.Then f E IX, b, c], a contradiction.

CASE 2 .
f(b) e analogous to Case 1.