Magill-type Theorems for Mappings

Magill's and Rayburn's theorems on the homeomorphism of Stone-ˇ Cech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings.

We note that results concerning the extension of Theorem 1.1 to mappings are also contained in [5], but they are different from ours (see the remark at the end of this paper).

Preliminaries.
Throughout this paper, space will mean a topological space and mapping will mean a continuous function.Terms and undefined concepts are used as in [4].In this section, we recall some definitions and results from [2].Some additional notions concerning fibrewise general topology (FGT) can be found in [9,10].Definition 2.1.Let X, Y , Z be spaces and λ : X → Y , µ : X → Z mappings.We say that λ is equivalent to µ and we will write λ ≡ µ if there exists some homeomorphism h : Y → Z such that µ = h • λ.
Evidently, the homeomorphism h is unique.We will identify equivalent mappings, and so we can consider the set Ꮿ(X) of all the continuous maps from a fixed space X onto other spaces.Definition 2.2.Let λ, µ ∈ Ꮿ(X).We say that λ follows µ and we will write λ ≥ µ if there exists some continuous mapping h : Y → Z such that µ = h • λ.
We will suppose from this moment that X is a Hausdorff space and that ᏼ(X) denotes the poset (as a subposet of Ꮿ(X)) of all perfect onto mappings of X.Clearly, λ(X) is Hausdorff for any λ ∈ ᏼ(X).Definition 2.4.A mapping λ ∈ ᏼ(X) is called a dual point if it is simple and |λ −1 ({t λ })| = 2.
Let Ᏸ = Ᏸ(X) denote the set of all dual points of ᏼ(X) and Ᏺ(X) = {λ ∈ ᏼ(X) : λ is finite simple}∪{id X }.Definition 2.5.A family Ᏺ ⊂ Ᏸ is said to be a 3-vertex family if for any distinct α, β ∈ Ᏺ there exists some γ ∈ Ᏸ \ Ᏺ such that γ > inf{α, β}.Definition 2.6.A 3-vertex family Ᏺ ⊂ Ᏸ is called a point family if it is maximal (i.e., if there is no 3-vertex family properly containing Ᏺ). Lemma 2.7.If Ᏺ is a 3-vertex family consisting of more than one element, then the set X Ᏺ = {λ −1 ({t λ }) : λ ∈ Ᏺ} is a single point (which will be denoted by J Ᏽ (Ᏺ)). (2.1) In [2], dual points are characterized only by means of the order in Ᏽ.It follows from this that if, for Hausdorff spaces X j , we have Ᏺ(X j ) ⊂ Ᏽ j ⊂ ᏼ(X j ) with j = 1, 2 and i : We recall [2] that a one-to-one mapping f : X → Y between Hausdorff spaces is called a k-homeomorphism if f is continuous on compact subsets of X and its inverse f −1 is continuous on compact subsets of Y .Clearly, a k-homeomorphism between k-spaces is a homeomorphism.Theorem 2.10.Let X j be a Hausdorff space and ᏼ(X j ) the set of all perfect onto mappings of X j (with j = 1, 2).If X 1 and X 2 are k-homeomorphic and they are k-spaces, then ᏼ(X 1 ) and ᏼ(X 2 ) (and so Ᏺ(X 1 ) and Ᏺ(X 2 )) are isomorphic.Let Ᏺ(X j ) ⊂ Ᏽ j ⊂ ᏼ(X j ) for j = 1, 2. If Ᏽ 1 and Ᏽ 2 are poset isomorphic, then X 1 and X 2 are k-homeomorphic, and if, additionally, X 1 and X 2 are k-spaces, then they are homeomorphic.More precisely, if i : Now, let X be a Tychonoff space and let K(X) denote the poset of all Hausdorff compactifications of X (see, e.g., [3]).
For any cX, dX ∈ K(X) such that cX < dX, let 3) be the canonical map (i.e., it is continuous and π dc|X = id X ).Then π −1 dc (cX \ X) = dX \ X and the mapping is perfect and onto, that is, r π dc ∈ ᏼ(dX \ X).
Fix eX ∈ K(X) and let In [2], the function was defined by σ eX (cX) = r π ec and the following lemma was proved.
Lemma 2.11.σ eX is an isomorphism of the posets K(eX) and (2.7) The following two lemmas were also proved in [2].

On the homeomorphisms of two pairs of spaces
Definition 3.1.A space X and a closed subset A are called a pair of spaces and denoted by (X, A).Definition 3.2.Suppose (X, A) and (Y , B) are pairs of spaces, where X, Y are Hausdorff.Then a homeomorphism (a k-homeomorphism) h : Let X be a Hausdorff space and A a closed subset of X.
that is, res XA (λ) is the corestriction of λ to A. Evidently, res XA (ᏼ(X)) ⊂ ᏼ(A) and res XA : ᏼ(X) → ᏼ(A) is monotonous.It is not difficult to prove the following lemma.
Let Ᏺ(X) ⊂ Ᏽ(X) ⊂ ᏼ(X) and Ᏺ(A) ⊂ Ᏽ(A) ⊂ ᏼ(A).Then, clearly, ) and so If, additionally, X and Y are k-spaces, then h i XY and h i AB are homeomorphisms, and so h i XY is a homeomorphism of (X, A) onto (Y , B).
Proof.First, let min{|A|, |B|} ≥ 3. Let x ∈ A. Then, by Theorem 2.10, h i XY and h i AB are k-homeomorphisms and 2) and since i XY is a poset isomorphism There is a unique dual point λ ∈ Ᏽ(X) (3.10)

Extensions of Magill's and Rayburn's theorems to mappings
It is not difficult to prove the following lemma.Lemma 4.2.For mappings f j : X j → Y (with j = 1, 2) and for (k-) homeomorphism h : X 1 → X 2 of spaces X 1 and X 2 , the following conditions are equivalent: Given a Tychonoff space X and a closed subset A of X, we may define a function Consequently, (eA = kres XA (eX)) > (cA = kres XA (cX)) and π ecA = π ecX : eA → cA.Thus, kres XA is monotone and r π ecA = r π ecX : We have then proven the following lemma.
Lemma 4.3.If eX ∈ K(X) and eA = kres XA (eX), then Let f : X → Y be a mapping to a Tychonoff space Y and let βf + : βX → βY be the (usual) continuous extension of f over the Stone-Cěch compactifications βX, β f X = (βf + ) −1 (Y ), and βf = βf if the space X is normal and f is a WZ-mapping.It is known [6] that every Z-mapping is a WZ-mapping.Theorem 4.4.Let X j be a Tychonoff space, let Y be a compact Hausdorff space, let f j : X j → Y be a WZ-mapping, let eX j be a Hausdorff compactification of X j , and let ef j : eX j → Y be a continuous extension of f j (thus, ef j is a compactification of f j ) for j = 1, 2. Let also X jy = f −1 j ({y}), eX jy = cl eX j (X jy ) (i.e., eX jy = kres X j X jy (eX j )) for j = 1, 2, and suppose that there exist poset isomorphisms i : K(eX 1 ) → K(eX 2 ) and i y : K(eX 1y ) → K(eX 2y ) such that Then the remainders ef j \f j def = ef j : additionally, X 1 and X 2 are k-absolute spaces, then the remainders ef 1 \ f 1 and ef 2 \ f 2 are homeomorphic.
Proof.It is sufficient to apply Theorem 4.4 to (the simplest) mappings f j of X j to the single point space Y for j = 1, 2.
In particular, in Corollary 4.5, for eX j = βX j (j = 1, 2), we have Rayburn's Theorem 1.3.Thus, Theorem 4.4 is a generalization of this theorem to mappings.Theorem 4.6.Let X j be a locally compact Tychonoff space, let Y be a compact Hausdorff space, let f j : X j → Y be a WZ-mapping, let eX j be a Hausdorff compactification of X j , and let ef j : eX j → Y be a continuous extension of f j for j = 1, 2. Let also X jy = f −1 j ({y}), eX jy = cl eX j (X jy ) (i.e., eX jy = kres X j X jy (eX j )) (for j = 1, 2).Then the remainders ef j \ f j = ef j : eX j \ X j → Y of ef j for j = 1, 2 are homeomorphic if and only if there exist poset isomorphisms i : K(eX 1 ) → K(eX 2 ) and i y : K(eX 1y ) → K(eX 2y ) such that (4.4) holds.
Corollary 4.7 [2].Let X 1 , X 2 be locally compact Tychonoff spaces and let eX 1 , eX 2 be Hausdorff compactifications of X 1 and X 2 , respectively.Then the remainders eX 1 \X 1 and eX 2 \ X 2 are homeomorphic if and only if K(eX 1 ) and K(eX 2 ) are poset isomorphic.
Proof.It is sufficient to apply Theorem 4.6 to the simplest mappings f j of X j and ef j of eX j to the single point space Y for j = 1, 2.
In particular, when in Corollary 4.7, eX j = βX j for j = 1, 2, we have Magill's theorem from [8].Thus, Theorem 4.6 is a generalization of this theorem to mappings.

Reformulations of results obtained above and some examples.
Some readers may find that Theorems 4.4 and 4.6 do not sound very natural.The reformulations, in the framework of FGT, sound better to us.
We will start with some definitions and results of FGT.
A mapping is called compact if it is perfect.
The following is evident.
Lemma 5.1.For a compact Hausdorff space Y , a mapping f : X → Y is compact if and only if X is compact.Definition 5.2.A mapping f : X → Y is said to be T 0 [11] if for every x, x ∈ X such that x = x and f (x) = f (x ), there exists a neighbourhood of x in X which does not contain x or a neighbourhood of x in X not containing x. Definition 5.3.A mapping f : X → Y is said to be completely regular [11] if for every closed set F of X and x ∈ X \F , there exist a neighbourhood O of f (x) in Y and a continuous function ϕ : It is not difficult to prove the following lemma.Lemma 5.4 [11].For a Tychonoff space Y , a mapping f : , X is dense in e f X, and e f X| X = f (more precisely, if some embedding e : X → e f X is fixed so that e(X) is dense in e f X and f = ef • e, but usually, X and e(X) are identified by means of e).
Throughout the rest of the paper, we fix a space Y and we will consider only Tychonoff mappings to Y and their Tychonoff compactifications.Definition 5.6.A mapping λ : In this case, one says that we have a canonical morphism λ : df → cf (and we write that df > cf ).
It is not difficult to prove that df and cf are homeomorphic if and only if df > cf and cf > df (see, e.g., [1]).
It is proved in [11] (see also [1]) that all compactifications of a mapping to Y form a set up to canonical homeomorphisms.This set will be denoted by T K(f ).Evidently, with respect to the just defined relation >, T K(f ) is a poset.
In [11], it is also proved that there exists the maximal element βf : β f X → Y in T K(f ) and that, if Y is Tychonoff, βf may be obtained in the following way.
By Lemma 5.4, X is Tychonoff.Hence, there exists the unique continuous extension βf + : βX → βY of f .Then For a compactification ef : e f X → Y of a mapping f : X → Y , the mapping ef \ f = ef : e f X \ X → Y is called the remainder of ef .
The following proposition is proved in [5].

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 .6) By (4.3), for j = 1, 2, we have (see the diagram) res R ej R ejy •σ eX j = σ eX jy • kres X j X jy | K(eX j ) .

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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