The main purpose of this paper is to study the fixed point property of
non-metric tree-like continua. Using the inverse systems method, it is proved that if X is a non-metric tree-like continuum and if f:X­→X is a mapping which sends each arc component into itself,
then f has the fixed point property

I. Lončar

The main purpose of this paper is to study the fixed point property of

non-metric tree-like continua. Using the inverse systems method, it is proved that if X is a non-metric tree-like continuum and if f:X­→X is a mapping which sends each arc component into itself,

then f has the fixed point property

Lončar, I.

Hrčak - Portal of scientific journals of Croatia

Mathematical Communications 10(2005), 15-21 15The fixed point property for arc componentpreserving mappings of non-metric tree-likecontinuaIvan Loncˇar∗Abstract. The main purpose of this paper is to study the fixed pointproperty of non-metric tree-like continua. Using the inverse systemsmethod, it is proved that if X is a non-metric tree-like continuum andif f : X → X is a mapping which sends each arc component into itself,then f has the fixed point property.Key words: continuum, fixed point property, inverse systemAMS subject classifications: Primary 54H25, 54F15; Secondary54B35Received April 12, 2004 Accepted February 10, 20051. IntroductionAll spaces in this paper are compact Hausdorff and all mappings are continuous.The weight of a space X is denoted by w(X). The cardinality of a set A is denotedby card(A). We shall use the notion of an inverse system as in [2, pp. 135-142]. Aninverse system is denoted by X = {Xa, pab, A}.Let A be a partially ordered directed set. We say that a subset A1 ⊂ Amajorates[1, p. 9] another subset A2 ⊂ A if for each element a2 ∈ A2 there exists an elementa1 ∈ A1 such that a1 ≥ a2. A subset which majorates A is called cofinal in A. Asubset of A is said to be a chain if its every two elements are comparable. Thesymbol supB, where B ⊂ A, denotes the lower upper bound of B (if such anelement exists in A). Let τ ≥ ℵ0 be a cardinal number. A subset B of A is said tobe τ -closed in A if for each chain C ⊂ B, with card(B) ≤ τ, we have supC ∈ B,whenever the element supC exists in A. Finally, a directed set A is said to beτ -complete if for each chain C of A of elements of A with card(C) ≤ τ , there existsan element supC in A.Suppose that we have two inverse systemsX = {Xa, pab, A} andY = {Yb, qbc, B}.Amorphism of the system X into the system Y [1, p. 15] is a family {ϕ, {fb : b ∈ B}}∗Faculty of Organization and Informatics Varazˇdin, University of Zagreb, Pavlinska 2, HR-42 000 Varazˇdin, e-mail: ivan.loncar1@vz.tel.hr16 I. Loncˇarconsisting of a nondecreasing function ϕ : B → A such that ϕ(B) is cofinal in A,and of maps fb : Xϕ(b) → Yb defined for all b ∈ B such that the followingXϕ(b)pϕ(b)ϕ(c)←− Xϕ(c)↓ fb ↓ fcYbqbc←− Yc(1)diagram commutes. Any morphism {ϕ, {fb : b ∈ B}} : X → Y induces a map,called the limit map of the morphismlim{ϕ, {fb : b ∈ B}} : limX→ limY (2)In the present paper we deal with the inverse systems defined on the sameindexing set A. In this case, the map ϕ : A→ A is taken to be the identity and weuse the following notation {fa : Xa → Ya; a ∈ A} : X→ Y.We say that an inverse system X = {Xa, pab, A} is factorizing [1, p. 17] iffor each real-valued mapping f : limX→ R there exist an a ∈ A and a mappingfa : Xa → R such that f = fapa.An inverse system X = {Xa, pab, A} is said to be σ-directed if for each sequencea1, a2, ..., ak, ... of the members of A there is an a ∈ A such that a ≥ ak for each k∈ N.Lemma 1. [1, Corollary 1.3.2, p. 18]. If X = {Xa, pab, A} is a σ-directedinverse system of compact spaces with surjective bonding mappings, then it is fac-torizing.An inverse system X = {Xa, pab, A} is said to be τ-continuous [1, p. 19]if for each chain B in A with card(B) < τ and supB = b, the diagonal product∆ {pab : a ∈ B}maps the spaceXb homeomorphically into the space lim{Xa, pab, B}.An inverse system X = {Xa, pab, A} is said to be a τ-system [1, p. 19] if:a) w(Xa) ≤ τ for every a ∈ A,b) The system X = {Xa, pab, A} is τ -continuous,c) The indexing set A is τ -complete.A σ-system is a τ -system, where τ = ℵ0. The following theorem is called theSpectral Theorem [1, p. 19].Theorem 1. [1, Theorem 1.3.4, p. 19]. If a τ-system X = {Xa, pab, A} withsurjective limit projections is factorizing, then each map of its limit space into thelimit space of another τ-system Y = {Ya, qab, A} is induced by a morphism ofcofinal and τ-closed subsystems. If two factorizing τ-systems with surjective limitprojections and the same indexing set have homeomorphic limit spaces, then theycontain isomorphic cofinal and τ-closed subsystems.Let us remark that the requirement of surjectivity of the limit projections ofsystems in Theorem 1 is essential [1, p. 21].The fixed point property ... 172. The fixed point property of non-metric continuaA fixed point of a function f : X → X is a point p ∈ X such that f(p) = p. A spaceX is said to have the fixed point property provided that every mapping f : X → Xhas a fixed point.The key theorem is the following.Theorem 2. Let X = {Xa, pab, A} be a σ-system of compact spaces withlimit X and onto projections pa : X → Xa. Let {fa : Xa → Xa} : X → X be amorphism. Then the induced mapping f = lim {fa} : X → X has a fixed point ifand only if each mapping fa : Xa → Xa, a ∈ A, has a fixed point.Proof. The if part. Let Fa, a ∈ A, be the set of fixed points of the mapping fa.Claim 1. Every set Fa is closed. This is a consequence of the following theorem[2, Theorem 1.5.4., p. 59]: For any pair f,g of mappings of a space X into aHausdorff space Y, the set{x ∈ X : f(x) = g(x)} (3)is closed in X.It suffices to set g(x) = x and Y = X .Claim 2. If a ≤ b, then pab(Fb) ⊂ Fa. Let xb be any point of Fb. Fromthe commutativity of diagram (1) it follows pab(fb(x)) = fa(pab(xb)). We havepab(xb) = fa(pab(xb)) since fb(x) = xb. This means that for the point y = pab(xb) ∈Xa we have y = fa(y), i.e., y ∈ Fa. We infer that pab(xb) ∈ Fa and pab(Fb) ⊂ Fa.Claim 3. F = {Fa, pab|Fb, A} is an inverse system of compact spaces with anon-empty limit F.Claim 4. The set F ⊂ X is the set of fixed points of the mapping f . Let x ∈ Fand let xa = pa(x), a ∈ A. Now, fa(xa) = xa since xa ∈ Fa. We infer that f(x) = xsince the morphism {fa : a ∈ A} induces f . The proof of the ”if” part is complete.The only if part. Suppose that the induced mapping f has a fixed point x. Letus prove that every mapping fa, a ∈ A, has a fixed point. Now we have fapa(x) =paf(x). From f(x) = (x) it follows fapa(x) = pa(x). We infer that pa(x) is a fixedpoint for fa. ✷As an immediate consequence of this theorem and the Spectral theorem 1 wehave the following result.Theorem 3. Let a non-metric continuum X be the inverse limit of an inverseσ-system X = {Xa, pab, A} such that each Xa has the fixed point property and eachbonding mapping pab is onto. Then X has the fixed point property.3. The fixed point property of the inverse limit space oftree-like continuaA continuum X with precisely two non-separating points is called a generalized arc.A simple n-od is the union of n generalized arcs A1O,A2O, ..., AαO, each twoof which have only the point O in common. The point O is called the vertex or thetop of the n-od.By a branch point of a compact space X we mean a point p of X which is thevertex of a simple triod lying in X . A point x ∈ X is said to be the end point of18 I. LoncˇarX if for each neighborhood U of x there exists a neighborhood V of x such thatV ⊂ U and card(Bd(V )) = 1.Let S be the set of all end points and all branch points of a continuum X . Anarc pq in X is called a free arc in X if pq ∩ S = {p, q}.A continuum is a graph if it is the union of a finite number of metric free arcs.A tree is an acyclic graph.A continuum X is tree-like (arc-like) if for each open cover U of X , there is atree (arc) XU and a U-mapping fU : X → XU (the inverse image of each point iscontained in a member of U).Every tree-like continuum is hereditarily unicoherent. Every arc-like continuumis tree-like.Let Y X be the set of all mappings of X to Y . If Y is a metric space with ametric d, then on the set Y X one can define a metric d̂ by lettingd̂(f, g) = supx∈Xd (f(x), g(x)) . (4)Proposition 1. Let X be any tree-like continuum, let P be a polyhedron witha given metric d, r > 0 a real number and f : X → P a mapping. Then there exista tree Q, a mapping g : X → Q and a mapping p : Q→ P such that g(X) = Q andd̂(f, pg) ≤ r.Proof. Let K be a triangulation of P of mesh not greater than r/2. Let ai bethe vertices of K, and let St ai be the open star of K around the vertex ai. Hence,{St ai} is an open covering for P , and so is U = {f−1(St ai)} for X . There exista tree Q and a mapping g : X → Q such that g is a U-mapping and g(X) = Q.There exists a triangulation L of Q with vertices bj such that the cover V = {g−1(Stbj)} refines the cover U . Let x be a point of X and let s be a simplex of Q withvertices bj1 , ..., bjk containing g(x). This means that{g−1(St bj1), ..., g−1(St bjk)}is a collection of some g−1(St bj) containing x. It follows that g−1(St bj1)∩ ... ∩g−1(St bjk) = ∅. We infer that St bj1∩ ... ∩ St bjk = ∅. Let p : Q → P be asimplicial mapping sending each vertex bj of Q into a vertex ai having the propertythat g−1(St bi) ⊂ f−1(St ai). It remains to prove that d(f, pg) ≤ r. Now, for eachg−1(St bij ) we have some f−1(St aij ) with g−1(St bij ) ⊂ f−1(St aij ). From g−1(Stbj1)∩ ... ∩ g−1(St bjk) = ∅ it follows that f−1(St bj1)∩ ... ∩ f−1(St bjk) = ∅, i.e.,that there exists a simplex σ of K with vertices bj1 , ..., bjk such that f(x) ∈ St σ.Clearly, pg(x) ∈ St σ. Finally, d̂(f, pg) ≤ r. ✷Proposition 2. If X = {Xa, pab, A} is an inverse system of tree-like continuaand if pab are onto mappings, then the limit X = limX is a tree-like continuum.Proof. Let U = {U1, ..., Un} be an open covering of X . There exist an a ∈A and an open covering Ua = {U1a, ..., Uka} such that{p−1a (U1a), ..., p−1a (Uka)}refines the covering U . There exist a tree Ta and a Ua-mapping fUa : Xa → Tasince Xa is tree-like. It is clear that fUapa : X → Ta is a U-mapping. Hence, X istree-like. ✷Proposition 3. If X is a tree-like continuum, Q a tree and f : X → Q amapping, then f(X) is also a tree.Proof. This follows from the fact that a subcontinuum of a tree is a tree. ✷Now we shall prove an expanding theorem of tree-like continua into inverse σ-systems of metric tree-like continua.The fixed point property ... 19Theorem 4. If X is a non-metric tree-like continuum, then there exists a σ-system Xσ = {X∆, P∆Γ, Aσ} of metric tree-like continua X∆ and onto mappingsP∆Γ such that X is homeomorphic to limXσ.Proof. Let us observe that Propositions 1-3 are conditions (A)-(C) in [4, p.220]. Then from Mardesˇic´’s General Expansion Theorem [4, Theorem 2] it followsthat there exists an inverse systemX = {Xa, pab, A} of metric tree-like continuaXaand onto bonding mappings pab such that X is homeomorphic to limX. It remainsto prove that there exists such σ-system. The proof is broken down into severalsteps.Step 1. For each subset ∆0 of (A,≤) we define sets ∆n, n = 0, 1, ..., by theinductive rule ∆n+1 = ∆n⋃ {m(x, y) : x, y ∈ ∆n}, where m(x, y) is a member ofA such that x, y ≤ m(x, y). Let ∆ = ⋃{∆n : n ∈ N}. It is clear that card(∆) =card(∆0). Moreover, ∆ is directed by ≤. For each directed set (A,≤) we defineAσ = {∆ : ∅ = ∆ ⊂ A, card(∆) ≤ ℵ0 and ∆ is directed by ≤}. (5)Step 2. If A is a directed set, then Aσ is σ-directed and σ-complete. Let {∆1,∆2, ..., ∆n, ...} be a countable subset of Aσ. Then ∆0 = ∪{∆1, ∆2, ..., ∆n, ...}is a countable subset of Aσ. Define sets ∆n, n = 0, 1, ..., by the inductive rule∆n+1 = ∆n⋃ {m(x, y) : x, y ∈ ∆n}, where m(x, y) is a member of A such thatx, y ≤ m(x, y). Let ∆ = ⋃{∆n: n ∈ N}. It is clear that card(∆) = card(∆0). Thismeans that ∆ is countable. Moreover, ∆ ⊇ ∆i, i ∈ N. Hence, Aσ is σ-directed.Let us prove that Aσ is σ-complete. Let ∆1 ⊂ ∆2 ⊂ ...⊂ ∆n ⊂ ... be a countablechain in Aσ. Then ∆ = ∪{∆i : i ∈ N} is a countable and directed subset of A, i.e.,∆ ∈ Aσ. It is clear that ∆ ⊇ ∆i, i ∈ N. Moreover, for each Γ ∈ Aσ with propertyΓ ⊇ ∆i, i ∈ N, we have Γ ⊇ ∆. Hence ∆ = sup{∆i : i ∈ N}. This means that Aσ isσ-complete.Step 3. If ∆ ∈ Aσ, let X∆ = {Xb, pbb′ , ∆} and X∆ = limX∆. If ∆, Γ ∈ Aσand ∆ ⊆ Γ, let P∆Γ: XΓ → X∆ denote the map induced by the projectionspΓδ : XΓ → Xδ, δ ∈ ∆, of the inverse system XΓ.Step 4. If X = {Xa, pab, A} is an inverse system, then Xσ = {X∆, P∆Γ,Aσ} is a σ-directed and σ-complete inverse system such that limX and limXσ arehomeomorphic. Each thread x = (xa : a ∈ A) induces a thread (xa : a ∈ ∆) for each∆ ∈ Aσ, i.e., the point x∆ ∈ X∆. This means that we have a mapping H : limX→ limXσ such that H(x) = (x∆ : ∆ ∈ Aσ). It is obvious that H is continuous and1-1. The mapping H is onto since the collection of threads {x∆ : ∆ ∈ Aσ} inducesthe thread in X. We infer that H is a homeomorphism since limX is compact.Step 5. Every X∆ is a metric tree-like continuum. Apply Proposition 2.Step 6. Every projection P∆ : limXσ → X∆ is onto. This follows from theassumption that the bonding mappings pab are surjective.Finally, Xσ = {X∆, P∆Γ, Aσ} is the desired σ-system. ✷By a similar proof we obtain the following theorem.Theorem 5. If X is a non-metric arc-like continuum, then there exists a σ-system Xσ = {X∆, P∆Γ, Aσ} of metric arc-like continua X∆ and onto mappingsP∆Γ such that X is homeomorphic to limXσ.From [4, Theorem 1], [4, Theorem 2] and [4, Corollary 1] we obtain the followingwell known result [5, Theorem 2.13, p. 24].20 I. LoncˇarTheorem 6. Each metrizable tree-like (arc-like) continuum is homeomorphicwith the inverse limit of an inverse sequence of trees (arcs).Now we shall investigate the fixed point property of non-metric tree-like con-tinua. Let us recall the following known result.Theorem 7. [3, Theorem 1.2]. Suppose f is a map of a tree-like metriccontinuum M that sends each arc component of M into itself. Then f has a fixedpoint.A map f : X → X is a deformation if there exists a map H : X × [0, 1] → Xonto X such that H(p, 0) = p and H(p, 1) = f(p) for each point p ∈ X . We say thata map f : X → X is a generalized deformation if there exist a generalized arc L(with end points 0 and 1) and a map H : X×L→ X onto X such that H(p, 0) = pand H(p, 1) = f(p) for each point p ∈ X . Since (generalized) deformations sendarc-components into themselves, we have [3, Corollary 1.3].Corollary 1. Every metric tree-like continuum has the fixed point property fordeformations.For non-metric tree-like continua we shall prove the following theorem.Theorem 8. Let X = {Xa, pab, A} be an inverse σ-system of metric tree-likecontinua with onto bonding mappings and with limit X. If f : X → X is a mappingwhich sends each arc component into itself, then f has a fixed point.Proof. By virtue of Theorem 1 there exist a cofinal subset B(f) of A andmappings fb : Xb → Yb, where b ∈ B(f), such that the mapping f is induced bya collection {fb : b ∈ B(f)}. From Theorems 2 and 7 it follows that it suffices toprove that each fb sends each arc component into itself. Let xb ∈ Xb. We have toprove that xb and fb(xb) lie in some arc component of Xb, i.e., there is an arc in Xbwith end points xb and fb(xb). There exists a point x ∈ X such that xb = pb(x).There exists a generalized arc L in X with end points x and f(x) since f sends eacharc component into itself. This means that fb(L) contains the points pb(x) = xband pbf(x) = fb(pb(x)) = fb(xb). We infer that there is an arc with end points xband fb(xb) since fb(L) is arcwise connected [6, p. 201, Theorem 9]. The proof iscomplete. ✷The non-metric analogue of Theorem 7 is the following result.Theorem 9. Let X be a non-metric continuum tree-like. If f : X → X is amapping which sends each arc component into itself, then f has a fixed point.Proof. Apply Theorems 4, 7 and 8. ✷Corollary 2. Every non-metric tree-like continuum has the fixed point propertyfor generalized deformations.Corollary 3. Let X be a non-metric tree-like arcwise connected continuum. Iff : X → X, then f has a fixed point.Acknowledgement. The author is very grateful to the referee for his help andvaluable suggestions.References[1] A.Chigogidze, Inverse Spectra, Elsevier, Amsterdam, 1996.[2] R.Engelking, General Topology, PWN, Warszawa, 1977.The fixed point property ... 21[3] C. L.Hagopian, The fixed point property for deformations of tree-like con-tinua, Fund. Math. 155(1998), 161-176.[4] S.Mardesˇic´, Chainable continua and inverse limits, Glas. Mat. Fiz. i Astr.14(1959), 219-232.[5] S. B.Nadler, Jr., Continuum Theory, Marcel Dekker, New York, 1992.[6] L.B.Treybig, Arcwise connectivity in continuous images of ordered com-pacta, Glas. Mat. Ser. III 21(41)(1986), 201 - 211.

The fixed point property for arc component preserving mappings of non-metric tree-like continua

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