© Hindawi Publishing Corp. ON FINITELY EQUIVALENT CONTINUA

For positive integers m and n, relations between (hereditary) m- and n-equivalence are studied, mostly for arc-like continua. Several structural and mapping problems concerning (hereditarily) finitely equivalent continua are formulated.

A continuum means a compact connected metric space.For a positive integer n, a continuum X is said to be n-equivalent provided that X contains exactly n topologically distinct subcontinua.A continuum X is said to be hereditarily n-equivalent provided that each nondegenerate subcontinuum of X is n-equivalent.If there exists a positive integer n such that X is n-equivalent, then X is said to be finitely equivalent.Thus, for n = 1, the concepts of "1equivalent" and "hereditarily 1-equivalent" coincide, and they mean the same as "hereditarily equivalent" in the sense considered, for example, by Cook in [2].
Observe the following statement.
Statement 1.Each subcontinuum of an n-equivalent continuum is mequivalent for some m ≤ n.Thus, each finitely equivalent continuum is hereditarily finitely equivalent.Some structural results concerning finitely equivalent continua are obtained by Nadler Jr. and Pierce in [9].They have shown that if a continuum X is (a) semi-locally connected at each of its noncut points, then it is finitely equivalent if and only if it is a graph; (b) aposyndetic at each of its noncut points and finitely equivalent, then it is a graph.Furthermore, in both cases (a) and (b), if X is n-equivalent, then each subcontinuum of X is a θ n+1 -continuum.Recall that Nadler Jr. and Pierce in [9, page 209] posed the following problem.Problem 2. Determine which graphs, or at least how many, are n-equivalent for each n.
The arc and the pseudo-arc are the only known 1-equivalent continua.In [10] Whyburn has shown that each planar 1-equivalent continuum is tree-like, and planarity assumption has been deleted after 40 years by Cook [2] who proved tree-likeness of any 1-equivalent continuum.But it is still not known whether or not the arc and the pseudo-arc are the only ones among 1-equivalent continua.
In contrast to 1-equivalent case, 2-equivalent continua need not be hereditarily 2-equivalent, a simple closed curve is 2-equivalent while not hereditarily 2-equivalent.The 2-equivalent continua were studied by Mahavier in [5] who proved that if a 2-equivalent continuum contains an arc, then it is a simple triod, a simple closed curve or irreducible, and that the only locally connected 2-equivalent continua are a simple triod and a simple closed curve.It is also shown that if X is a decomposable, not locally connected, 2-equivalent continuum containing an arc, then X is arc-like and it is the closure of a topological ray R such that the remainder cl(R) \ R is an end continuum of X.Furthermore, two examples of 2-equivalent continua are presented in [5]: the first, [5, Example 1, page 246], is a decomposable continuum X which is the closure of a ray R such that the remainder cl(R) \ R is homeomorphic to X; the second, [5, Example 2, page 247], is an arc-like hereditarily decomposable continuum containing no arc.
Looking for an example of a hereditarily 2-equivalent continuum note that the former example surely is not hereditarily 2-equivalent because it contains an arc.We analyze the latter one.
The continuum M constructed in [5, Example 2, page 247] does not contain any arc, and it contains a continuum N such that each subcontinuum of M is homeomorphic to M or to N, see [5, the paragraph following Lemma 3, page 249].Further, by its construction, N does contain continua homeomorphic to M (see [5, the final part of the proof, page 251]).Therefore, the following statement is established.In connection with the above theorem, the following problem can be posed.Theorem 5.For each hereditarily n-equivalent continuum X, that does not contain any arc, there exists an (n + 2)-equivalent continuum Y such that each of its subcontinua is homomorphic either to a subcontinuum of X or to Y , or to an arc.
Proof.Indeed, a compactification Y of a ray R having the continuum X as the remainder, that is, such that Since if M is arc-like and hereditarily decomposable, then so is any of compactifications Y of a ray having the continuum X as the remainder, we get the next result as a consequence of Theorem 5.

Corollary 6. If a continuum M satisfies conditions (a), (b), and (c) of Theorem 3 and is hereditarily n-equivalent, then any of compactifications of a ray having the continuum M as the remainder satisfies conditions (a) and (b)
of Theorem 3 and is (n + 2)-equivalent.
In [7], an uncountable family Ᏺ is constructed of compactifications of the ray with the remainder being the pseudo-arc.Statement 7.Each member X of the (uncountable) family Ᏺ constructed in [7] is an arc-like 3-equivalent continuum.Any subcontinuum of X is homeomorphic to an arc, to a pseudo-arc, or to the whole X.
A continuum X has the RNT-property (retractable onto near trees) provided that for each ε > 0, there exists a δ > 0 such that if a tree T is δ-near to X with respect to the Hausdorff distance, then there is an ε-retraction of X onto T , see [6, Definition 0].It is shown in [6,Theorem 5] that if a continuum X is a compactification of the ray R and X has the RNT-property, then the remainder cl(R) \ R ⊂ X = cl(R) is the pseudo-arc.Therefore, Theorem 5 implies the following proposition.Proposition 8.Each compactification X of the ray having the RNT-property is a 3-equivalent continuum.Each subcontinuum of X is homeomorphic to an arc, a pseudo-arc, or to the whole X.
Observe that M of Theorem 3 being an arc-like is hereditarily unicoherent, and being hereditarily decomposable, it is a λ-dendroid (containing no arc).Another (perhaps the first) example of a λ-dendroid, in fact, an arc-like, containing no arc, has been constructed by Janiszewski in 1912, [3] but his description was rather intuitive than precise.It would be interesting to investigate if that old example of Janiszewski is or is not n-equivalent (hereditarily n-equivalent) for some n.
The following problems can be considered as a program of a study in the area rather than particular questions.Sometimes a characterization of a class of spaces (or of spaces having a certain property) can be expressed in terms of containing some particular spaces.A classical illustration of this is a well-known characterization of nonplanar graphs by containing the two Kuratowski's graphs: K 5 and K 3,3 , see, for example, [8, Theorem 9.36, page 159].To be more precise, recall the following concept.Let Ꮽ be a class of spaces and let ᏼ be a property.Then ᏼ is said to be finite (or countable) in the class Ꮽ provided that there is a finite (or countable, respectively) set of members of Ꮽ such that a member X has the property ᏼ if and only if X contains a homeomorphic copy of some member of .The result of [7] mentioned above in Statement 7 shows that this is not the way of characterizing 3-equivalent continua.Namely, the existence of the family Ᏺ shows the following theorem.
Theorem 11.The property of being 3-equivalent is neither finite nor countable in the class of (a) all continua; (b) arc-like continua.
A mapping f : X → Y between continua X and Y is said to be (i) atomic provided that for each subcontinuum K of X, either f (K) is degenerate or f −1 (f (K)) = K; (ii) monotone provided that the inverse image of each subcontinuum of Y is connected; (iii) hereditarily monotone provided that for each subcontinuum K of X, the partial mapping f |K : K → f (K) is monotone.It is known that each atomic mapping is hereditarily monotone, see, for example, [4, (4.14), page 17].Since each arcwise connected 2-equivalent continuum is either a simple closed curve or a simple triod, see [5, Theorem 2, page 244], each semilocally connected 3-equivalent continuum is either a simple 4-od [8, Definition 9.8, page 143] (i.e., a letter X) or a letter H, see [9, page 209].And since these continua are preserved under atomic mappings (as it is easy to see), we conclude that atomic mappings preserve the property of being 2-equivalent and being 3-equivalent for locally connected continua.However, this is not an interesting result, because each atomic mapping of an arcwise connected continuum onto a nondegenerate continuum is a homeomorphism, see [4, (6.3), page 51].But the result cannot be extended to hereditarily monotone mappings, because a mapping that shrinks one arm of a simple triod to a point is hereditarily monotone and not atomic, and it maps a 2-equivalent continuum onto an arc that is 1-equivalent.On the other hand, if X is the 2equivalent continuum which is the closure of a ray R as described in [5, Example 1, page 246], then the mapping f : X → [0, 1], that shrinks the remainder cl(R) \ R to a point (and is a homeomorphism on R), is atomic and it maps 2-equivalent continuum X onto the 1-equivalent continuum [0, 1].Therefore, atomic mappings do not preserve the property of being a 2-equivalent continuum.In connection with these examples, the following question can be asked.
Question 12. Let a continuum X be n-equivalent and let a mapping f : X → Y be an atomic surjection.Must then Y be m-equivalent for some m ≤ n?
In general, we can pose the following problems.
Problems 13.What kinds of mappings between continua preserve the property of being: (a) n-equivalent?(b) hereditarily n-equivalent?(c) finitely equivalent?

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Theorem 3 .
The continuum M constructed in [5, Example 2, page 247] has the following properties: (a) M is an arc-like; (b) M is hereditarily decomposable; (c) M does not contain any arc; (d) M is hereditarily 2-equivalent.

Problem 4 .
Determine for what integers n ≥ 3, there exists a continuum M satisfying conditions (a), (b), and (c) of Theorem 3 and being hereditarily n-equivalent.The following results are consequences of [1, Theorem, page 35].

Problems 9 .Problem 10 .
For each positive integer n, characterize continua which are (a) n-equivalent; (b) hereditarily n-equivalent.Characterize continua which are finitely equivalent.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation