The freudenthal space for approximate systems of compacta and some applications

Publicacions Matem`atiques, Vol 39 (1995), 215–232. THE FREUDENTHAL SPACE FOR APPROXIMATE SYSTEMS OF COMPACTA AND SOME APPLICATIONS Ivan Lonˇ car Abstract In this paper we define a space σ(X) for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We stablish the following properties of this space: (1) The space σ(X) is a paracompact space, (2) Moreover, if Xis an approximate sequence of compact (metric) spaces, then σ(X) is a compact (metric) space (Lemma 2.4). We give the following applications of the space σ(X): (3) If Xis an approximate system of continua, then X= lim Xis a continuum (Theorem 3.1), (4) If Xis an approximate system of hereditarily unicoherent spaces, then X= lim X is hereditarily unicoherent (Theorem 3.6), (5) If Xis an approximate system of trees with monotone onto bonding mappings, then X= lim Xis a tree (Theorem 3.13). 1. Introduction Let Ube any covering of a space X. For any subset Yof Xwe define St(Y,U)=∪{U∈U:U∩Y=∅}. Similarly, we define St U={St(U, U):U∈U}. Inductively, for each positive integer n,St nU= St(Stn−1U), where St1U=StU. We say that a cover Vis a star refinement ofacoverUif the cover St Vis a refinement of U. An open cover Wof a space Xis normal [3, p. 379] if there exists a sequence W1,W2,... of open covers of the space Xsuch that W1=W and Wi+1 is a star refinement of Wifor i=1,2,... AT1space Xis paracompact iff each open cover of Xis normal [3, Theorem 5.1.12]. A T1space Xis normal iff each locally finite open cover of Xis normal [3, p. 379]. The set of all normal covers of Xis denoted by Cov(X). 216 I. Lonˇ car If U,V∈Cov(X) and Vrefines U, we write V≺U.Iff,g:Y→X are U-near mappings, i.e. if for any y∈Ythere exists U∈Uwith f(y), g(y)∈U, we write (f,g)≺U. Approximate inverse systems were introduced by S. Mardeˇsi´c and L. R. Rubin [11] for compacta and by S. Mardeˇsi´c and Watanabe [12] for general topological spaces. Definition 1.1. An approximate inverse system X={Xa,Ua,p ab,A} consists of the following data: A preordered set (A, ≤) which is directed and has no maximal element; for each a∈A, a topological space Xaand a normal covering Uaof Xa(called the mesh of Xa) and for each pair a≤bfrom A, a mapping pab :Xb→Xa. Moreover the following three conditions must be satisfied: (A1) The mappings pabpbc and pac are Ua-near, a≤b≤c, i.e. (pabpbc,p ac)≺U a. (A2) For each a∈Aand each normal cover U∈Cov(Xa) there is b≥a such that (pacpcd,p ad)≺U, whenever a≤b≤c≤d. (A3) For each a∈Aand each normal cover U∈Cov(Xa) there is b≥a such Uc≺p−1 ac (U)={p−1 ac (U):U∈U}for each c≥b. In the case of metric compact spaces we replace the normal coverings by real numbers [11]. If the spaces Xaare T1paracompact, then in the above definition one can use all open coverings on the spaces Xa,a∈A, since in this case each open cover is normal. Definition 1.2. An approximate map p={pa:a∈A}:X→Xa into an approximate inverse system X={Xa,Ua,p ab,A}is a collection of maps pa:X→Xa,a∈A, such that the following condition holds (AS) For any a∈Aand any U∈Cov(Xa) there is b≥asuch that (pacpc,p a)≺U for each c≥b. (See [12].) Definition 1.3. Let X={Xa,Ua,p ab,A}be an approximate inverse system and let p={pa:a∈A}:X→Xabe an approximate map. We say that pis a limit of Xprovided it has the following universal property [12, p. 592]: (UL) For any approximate map q={qa:a∈A}:Y→Xaof a space Ythere exists a unique map g:Y→Xsuch that pag=qafor any a∈A. Remark 1.4. If p:X→Xis a limit of X, then the space Xis determined up to a unique homeomorphism. Therefore, we often speak of the limit Xof Xand we write X= lim X. The Freudenthal space for approximate systems 217 Definition 1.5. Let X={Xa,Ua,p ab,A}be an approximate system. A point x=(xa)∈{Xa:a∈A}is called a thread of Xprovided it satisfies the following condition: (L) (∀a∈A)(∀U ∈Cov(Xa))(∃b≥a)(∀c≥b)pac(xc)∈st(xa,U). Remark 1.6. If Xais a T3.5space, then the sets st(xa,U), U∈ Cov(Xa), form a basis of the topology at the point xa. Therefore, for an approximate system of Tychonoff spaces condition (L) is equivalent to the following condition: (L)∗(∀a∈A) lim{pac(xc):c≥a}=xa. The following theorem shows that the set of threads is a limit of X. Theorem 1.7. Let X={Xa,Ua,p ab,A}be an approximate inverse system. Let X⊂Xabe the set of all threads of Xand let pa:X→Xa be the restriction pa=πa|Xof the projection πa:Xa→Xa,a∈A. Then p={pa:a∈A}X→Xis a limit of X. Proof: See [12, Theorem (1.14)]. The canonical limit of Xis the set of all threads of X[12, p. 593]. Theorem 1.8. For any approximate inverse system Xthe canonical limit lim Xis closed in Xa. Moreover, if all Xaare compact and non-empty, then lim Xis compact and non-empty. Proof: See the proof of Lemma (1.16) and Theorem (4.1) of [12]. Lemma 1.9. Let X={Xa,Ua,p ab,A}be an approximate inverse system of Tychonoff spaces, let Xbe the canonical limit of Xand let B⊆Abe a cofinal subset of A. Then the collection Bof all sets of the form p−1 b(Ub), where b∈Band Vb⊆Xbis open, is a basis of the topology for X. Proof: See [12, (1.18) Lemma]. Theorem 1.10. Let X={Xa,Ua,p ab,A}be an approximate inverse system of compact Hausdorff spaces with limit X. For each closed F⊆X we have F={p−1 a(pa(F)) : a∈A}. Proof: It is obvious that F⊆p−1 a(pa(F)) for each a∈A. Thus, F⊆ ∩{p−1 a(pa(F)) : a∈A}.Ifx/∈F, then, by Lemma 19 we infer that there exists an a∈Aand an open set Ua⊆Xasuch that x∈p−1 a(Va)⊆X−F. This means that pa(x)/∈pa(F) and x/∈p−1 a(pa(F)). 218 I. Lonˇ car 2. The Freudenthal space σ(X) The following construction is similar to the construction due to H. Freudenthal [4, p. 153] for usual inverse sequences. For any usual inverse system see [10]. Let X={Xa,Ua,p ab,A}be an approximate inverse system of compact Hausdorff spaces with limit Xand the projections pa:X= lim X→Xa. The Freudenthal space σ(X) associated to Xis the set (1) σ(X)=X{Xa:a∈A} where all Xaand their limit Xare considered as being disjoint sets [10], in which a topology is defined as follows. If Uais an open set in Xa, let (2) U∗ a={p−1 ab (Ua):b≥a}p−1 a(Ua). Now, we define a topology Ton σ(X) by a base [3, p. 27] Bwhich consists of all open sets Uain all Xaand all U∗ afor all open sets Ua⊆Xa,a∈A. Since the sets p−1 a(Ua) form a basis for X, it follows that Bisacoverof σ(X). By virtue of [3, p. 27] we need to prove that for each x∈σ(X) and each pair B,C∈Bwith x∈B∩Cthere is a D∈Bsuch that x∈D⊆B∩C. It suffices to prove this statement if Bis some U∗ a and Cis some U∗ b.Ifxis a point of Xc, then xis contained in a set p−1 ac (Ua)∩p−1 bc (Ub) which is open in Xcand thus belongs to B.Ifxis a point of X, then (3) z∈p−1 a(Ua)p−1 b(Ub) i.e., xa=pa(x)∈Ua, and xb=pb(x)∈Ub. Choose Va∈Cov(Xa), Vb∈Cov(Xb) such that (4) St(xa,Va)⊆Uaand St(xb,Vb)⊆Ub. Take Wa∈Cov(Xa), Wb∈Cov(Xb) such that St2Wa≺V a,St 2Wb≺ Vband c∈Asuch that c≥a,b, (A2) and (A3) hold for a,b,Wa,Wb and (L) holds for x,a,b,Wa,Wb. Put (5) Vc= St(xc,Uc). Since x∈p−1 c(Vc)⊆V∗ c, the proof will be complete if we show that (5.1) V∗ c⊆U∗ aU∗ b. The Freudenthal space for approximate systems 219 We first prove that (6) p−1 c(Vc)⊆p−1 a(Ua)p−1 b(Ub). Consider a point y=(ya)∈p−1 c(Vc). By (5) there is a U1∈U csuch that (7) xc,y c∈U1. By the choice of c(property (A3)) Uc≺p−1 ac (Wa) and Uc≺p−1 bc (Wb). This means that there is a W1∈W aand W2∈W bsuch that U1⊆ p−1 ac (W1) and U1⊆p−1 bc (W2). Thus, (7) implies (8) pac(xc),p ac(yc)∈W1and pbc(xc),p bc(yc)∈W2. By the choice of c(property (L)), there are W3∈W a,W4∈W bsuch that (9) xa,p ac(xc)∈W3and xb,p bc(xc)∈W4. Since y∈p−1 b(Ub)⊆X, there is a d≥csatisfying (L) for y,a,Waand for y,b,Wb. Thus, there exist a W5∈W a,W6∈W band U4∈U csuch that (10) pad(yd),y a∈W5and pbd(yd),y b∈W6 and (11) pcd(yd),y c∈U4. By the choice of c(property (A3)), Uc≺p−1 ac (Wa) and Uc≺p−1 bc (Wb). Hence, there exist a W7∈W aand W8∈W bsuch that U4⊆p−1 ac (W7) and U4⊆p−1 bc (W8). By (11) we have (12) pacpcd(yd),p ac(yc)∈W7and pbcpcd(yd),p bc(yc)∈W8. By the choice of c(property (A2)), we also have a W9∈W aand W10 ∈ Wbsuch that (13) pacpcd(yd),p ad(yd)∈W9and pbcpcd(yd),p bd(yd)∈W10. Now, (9), (8), (12), (13), (10), St2Wa≺V aand St2Wa≺V byield a V∈V aand a V ∈V bsuch that xa,ya∈W1∪W3∪W5∪W7∪W9⊆V and xb,yb∈W2∪W4∪W6∪W8∪W10 ⊆V. This and (4) imply 220 I. Lonˇ car pa(y)=ya∈St(xa,Va)⊆Uaand pb(y)=yb∈St(xb,Vb)⊆Ub. This means that y∈p−1 a(Ua)∩p−1 b(Ub), i.e., (6) is proved. It remains to prove (14) p−1 cd (Vc)⊆p−1 ad (Ua)p−1 bd (Ub)∀d≥c. Let zd∈p−1 cd (Vc). By (5) there is a U11 ∈U csuch that (15) xc,p cd(zd)∈U11. By the choice of c(property (A3)) there is a W11 ∈W aand a W12 ∈W b such that U11 ⊆p−1 ac (W11) and U11 ⊆p−1 bc (W12). Thus, (15) implies (16) pac(xc),p ac(pcd(zd)) ∈W11 and pbc(xc),p bc(pcd(zd)) ∈W12. By (A2) we infer there are W13 ∈W aand W14 ∈W bsuch that (17) pacpcd(zd),p ad(zd)∈W13 and pbcpcd(zd),p bd(zd)∈W14. From (9), (16) and (17) it follows xa,pad(zd)∈St Vaand xb,pbd(zd)∈ St Vb. By (4) pad(zd)∈Uaand pbd(zd)∈Ub. We infer that zd∈ p−1 ad (Ua)∩p−1 bd (Ub) and (14) is proved. Hence, we have x∈V∗ c⊆U∗ a∩U∗ b, i.e., (5.1) is proved. This means that Bis a basis for some topology T on σ(X). Now, we will prove that Tis a Hausdorff topology. Let x,ybe a pair of distinct points in σ(X). If x,y/∈lim X, then there exists a pair a, b∈Asuch that x∈Xa,y∈Xb.Ifa=b, then xand yhave disjoint neighborhoods since Xais a Hausdorff space. If a=b, then Xaand Xbare disjoint neighborhoods (in σ(X)) of xand yrespectively. Now, suppose that x∈lim Xand y/∈lim X. Let y∈Xbfor some b∈A. By virtue of Lemma 1.9 there is a c>band an open set Ucsuch that p−1 c(Uc) is a neighborhood of xin lim X. It is clear that Xband V∗ care disjoint neighborhoods of yand xin σ(X). Finally, let x,y∈lim X. Since lim Xis a Hausdorff space, there are open (in lim X) disjoint sets Uand Vsuch that x∈Uand y∈V. By virtue of Lemma 1.9 there exists a b∈Aand open sets Uband Vbsuch that x∈p−1 b(Ub)⊆Uand y∈p−1 b(Vb)⊆V. It follows that Uband Vbare disjoint since Uand V are disjoint. Hence, U∗ band V∗ bare disjoint. Thus, σ(X) is a Hausdorff space. Anet in a topological space X[3, p. 73] is an arbitrary function from a non-empty directed set Dto the space X. Nets will be denoted by N={xd:d∈D}. A point x∈Xis called a limit of a net N={xd: d∈D}if for every neighborhood Uof xthere is a d0∈Dsuch that xd∈Ufor each d≥d0. We say that the net Nconverges to x. A point x∈Xis called a cluster point of a net N={xd:d∈D}if for every neighborhood Uof xand every d0∈Dthere exists a d≥d0such that xd∈U. The Freudenthal space for approximate systems 221 Lemma 2.1. Let X={Xa,Ua,p ab,A}be an approximate inverse system of non-empty compact Hausdorff spaces with limit X. 1. If Ais a cofinal subset of A, then each family N={xa:xa∈ Xa,a∈A}is a net in σ(X)which has at least one cluster point x(in the topology T) such that x∈X⊆σ(X). 2. Each point x∈Xis the limit (in the topology T) of the net {pa(xa):a∈A}. Proof: For each a∈Awe consider the net Na={pab(xb):b∈A,b≥ a}. From the compactness of Xait follows that the set Caof all cluster points of Nais non-empty. Clearly, each Cais closed and compact in Xa. First, we prove (a) For each a∈AC ais a non-empty subset of pa(X). If we suppose that some ca∈Ca\pa(X), then caand pa(X) respectively, have disjoint neighborhoods Uand V. By virtue of the property (B3) [12, p. 606, 615] there is a b≥asuch that pac(Xc)⊆Vfor each c≥b, c∈A. This is impossible since there exists c≥bsuch that pac(xc)∈U (cais a cluster point of the net Na). From (a) it easily follows that (b) For each a∈Athe set p−1 a(Ca)is non-empty. By (b) there is ya∈p−1 a(Ca)⊆lim X,a∈A. Since lim Xis compact, there is a cluster point y∈lim Xof the net Y={ya:a∈A}. Let us prove (c) pa(y)∈Ca,a∈A. It suffices to prove that for each neighborhood Uaof pa(y) and each b0there exists a d≥b0such that pad(xd)∈Ua. Let Ube a normal cover of Xasuch that (18) St2(pa(y),U)⊆Ua. Let U1∈Ube such that pa(y)∈U1. Then p−1 a(U1) is a neighborhood of y. The set Bof all b∈Awith yb∈p−1 a(U1) is cofinal in Asince y is a cluster point of Y. By virtue of (AS) the set B⊆Bof all b∈B, b≥b0, such that (19) (pa,p abpb)≺U is cofinal in A. Similarly, by (A2), the set B ⊆Bof all b∈Bsuch that (20) (pac,p abpbc)≺U,c≥b 222 I. Lonˇ car is cofinal in A. Let b∈B. Then yb∈p−1 a(U1). Thus (21) pa(y),p a(yb)∈U1. By virtue of (19) it follows (22) pa(yb),p abpb(yb)∈U2∈U. This and (21) imply (23) pabpb(yb)∈St(pa(y),U). Now, pb(yb)∈Cbsince yb∈p−1 b(Cb). We infer that p−1 ab (St(pa(y),U)) is a neighborhood of pb(yb). Since pb(yb) is a cluster point of Nathere is ad≥b≥b0,d∈Asuch that pbd(xd)∈p−1 ab (St(pa(y),U)). This means that pab(pbd(xd)) ∈St(pa(y),U). Using (20), pad(xd)∈St2(pa(y),U). Thus, by (18) (24) pad(xd)∈Ua. We infer that pa(y)∈Ca, i.e., y∈p−1 a(Ca) for each a∈A. (d) The point yis a cluster point (in the topology T)ofN. This follows from (24) since xd∈p−1 ad (Ua). This means that for each neighborhood U∗ aof yand each b0∈Athere is a d≥b0,d∈A, such that xd∈U∗ a. The proof of Lemma 2.1 is complete since the second statement easily follows from the definition of the topology Ton σ(X). Lemma 2.2. Let X={Xa,Ua,p ab,A}be an approximate inverse system of compact Hausdorff spaces. If Uis a neighborhood of X= lim X in σ(X), then there exists a∈Asuch that Xb⊆Ufor each b≥a. Proof: Since Xis compact and since the sets (2) form a basis for the neighborhoods of the points of X, one can find {U∗ ai:i=1,... ,n}such that (25) V={U∗ ai:i=1,... ,n} and X⊆V⊆U. In order to complete the proof, it suffices to find an a∈A,a≥a1,... ,a nsuch that (26) Xa⊆V since then we have (27) Xb⊆V⊆U, b ≥a. Suppose that no a∈Asatisfies (26). This means that for each a∈A there is xa∈Xa−V. We obtain a net {xa:a∈A}in σ(X) which has no cluster point in V⊇X. This contradicts Lemma 2.1. The proof is complete. The Freudenthal space for approximate systems 223 Lemma 2.3. Let X={Xa,Ua,p ab,A}be an approximate inverse system of compact Hausdorff spaces. Then σ(X)is paracompact. Moreover, if Xis an approximate sequence, then σ(X)is compact. Proof: Let V={Vµ}be any cover of σ(X). Since Xis compact, there is a finite subcollection, consisting of sets Vµ(1),... ,V µ(n)which cover X. Let Vbe the union of this subcollection. By virtue of Lemma 2.2 there is an a∈Asuch that all Xb,b≥a, are in V. Let us recall that the set X∗ a=(∪{Xb:b≥a}∪Xis of type (2) with Ua=Xaand it is open in σ(X). Now consider the following collection Uof open sets of σ(X): take first the open sets X∗ a∩Vµ(1),... ,X∗ a∩Vµ(n)for members of U. Furthermore, for each b∈A−{c:c∈A, c ≥a}consider the open covering {Xb∩Vµ}of Xband take members of a finite subcovering as new members of U. This is possible since Xbis compact and open in σ(X). The family Uof open sets of σ(X) is a star-finite covering of σ(X) which refines the covering V. Moreover, Uis a locally finite refinement of V. The proof of paracompactness is complete. If Xis an approximate sequence, then we obtain a finite subcovering since the set A−{c:c∈A, c ≥a}is finite. The proof is complete. Theorem 2.4. Let X={Xn, n,p mn,N}be an approximate inverse sequence of compact metric spaces Xn. Then σ(X)is a compact metric space. Proof: Each space Xnhas a countable base Bn[3, 4.1.15 Theorem]. It follows that the family B∗={U∗:U∈B n:n∈N}is countable. It is obvious that the union B={Bn:n∈N}∪B ∗is a countable base for topology T.Thusσ(X) is metrizable [3, p. 351]. We close this section with the following theorem which is similar to the theorem for usual inverse systems of compact Hausdorff spaces due to S. Mardeˇsi´c[10, Theorem 4] (see Theorem 4.2 of [12]). Theorem 2.5. Let X={Xa,Ua,p ab,A}be an approximate inverse system of compact Hausdorff spaces and let f:X→Rbe a mapping of their limit into a simplicial complex. Then there exists an a∈Asuch that for each b≥aone can define a mapping fb:Xb→Rwith the property that fbpbis homotopic to f. 3. Applications In this section we give some applications of the space σ(X). We start with 230 I. Lonˇ car Lim{pa(I(y,xµ)) : a∈A}. Since each pa(I(y,xµ)) contains I(ya,x µ a), we infer that Kµ⊆I(y,xµ). Finally, we have Kµ=I(y,xµ). Step 3.2. For each a∈Aand each µ∈Mwe have pa(Kµ)=I(ya,x µ a). Clearly, pa(Kµ)⊇I(ya,x µ a). Suppose that there is an a∈Aand a point za∈pa(Kµ)\I(ya,x µ a). This means that there are disjoint open sets Uaand Vasuch that za∈Vaand I(ya,x µ a)⊆Ua. From the local connectedness of Xait follows that there is an open and connected set Wasuch that I(ya,x µ a)⊆ClWa⊆Ua. From the definition of thread it follows that there is a b∈Asuch that pac(yc) and pac(xµ c) are in Wafor each c≥b. This means that pac(I(yc,x µ c)) ⊆ClWasince pac(I(yc,x µ c)) is irreducible between pac(xc) and pac(xµ c) (see Lemma 3.11). It follows that U∗ ais a neighborhood of a point z∈K,pa(z)=za, such that U∗ a∩I(yc,x µ c)=∅. This means that z/∈Ls{I(ya,x µ a):a∈A}=Kµ. This is impossible since z∈Kµ. By Theorem 1.10 it follows that Kµ= ∩{p−1 a(I(ya,x µ a)) : a∈A}. Similarly, we have K=∩{p−1 a(I(ya,x a)) : a∈A}, where K=Ls{I(ya,x a):a∈A}. Step 3.3. Ls{Kµ:µ∈M}=Ls{I(y,xµ):µ∈M}=I(y,x). It is obvious that Ls{I(y,xµ):µ∈M}⊇I(y,x) since Ls{I(y,xµ): µ∈M}contains xand yand I(y,x) is irreducible between xand y.Now we prove that Ls{I(y,xµ):µ∈M}⊆I(y,x). Let zbe any point in X−I(y,x). By virtue of the definition of a base in X, there is an a∈A such that pa(z)=za/∈pa(I(y,x)) = (by Steps 3.1 and 3.2) I(ya,x a). This means that there is a neighborhood Uaof zaand a neighborhood Va of pa(I(y,x)) such that Ua∩Va=∅. By Step 3.2 pa(I(y,x)) = I(ya,x a). Since I(ya,x a) = Lim{I(ya,x µ a):µ∈M}we infer that there is a µ0∈M such that, for each µ≥µ0,Uaand I(ya,x µ a) are disjoint. From 3.2 it follows that p−1 a(Ua) and I(y,xµ) are disjoint. Since p−1 a(Ua)isa neighborhood of z, we infer that z/∈Ls{I(y,xµ):µ∈M}. Thus, Ls{I(y,xµ):µ∈M}=I(y,x) and 3.3 is proved. Step 3.4. I(y,x) = Lim{I(y,xµ):µ∈M}. Apply Step 3.3 and Lemma 3.12. By virtue of Lemma 3.10 and Step 3.4 it follows that Xis smooth at y. We infer that Xis smooth in any of its point y. This means that X is locally connected. The proof of Theorem 3.13 is complete. A Hausdorff continuum Xwith precisely two nonseparating points is called a generalized arc. A continuum Xis said to be an arc if Xis a metrizable generalized arc. A tree Xis a generalized arc if and only if Xis atriodic. The Freudenthal space for approximate systems 231 Theorem 3.14. Let X={Xa,Ua,p ab,A}be an approximate inverse system of generalized arcs. Then X= lim Xis atriodic. Proof: Suppose that Tis a subcontinuum of Xwhich is a triod. This means that Tis the sum of three generalized arcs Cx,Cy, and Cz, such that the common part of each two of them is the common part of all three of them and is a point. Let x∈Cx−(Cy∪Cz), y∈Cy−(Cx∪Cz), z∈Cz−(Cx∪Cy) and t=Cx∩Cy∩Cz. By virtue of the definition of a basis in X, there exist a∈Aand open sets Vx,Vy,Vzof Xawhich are pairwise mutually exclusive and which contain xa,ya,za, respectively, so that p−1 a(Vx)Cy=∅=p−1 a(Vx)Cz, p−1 a(Vy)Cx=∅=p−1 a(Vy)Cz, p−1 a(Vz)Cy=∅=p−1 a(Vz)Cx. Now, one of xa,yaor zalies between taand one of xa,yaor za. Suppose that ta≺xa≺ya. Then pa(Cy) intersects taand yaand hence xa, but pa(Cy) does not intersect Vx. This is a contradiction. So Xcontains no triod. Theorem 3.15. Let X={Xa,Ua,p ab,A}be an approximate inverse system of generalized arcs with limit X. If the bonding mappings are monotone and onto, then Xis a generalized arc. Proof: By virtue of Theorem 3.13 Xis a tree. From 3.14 it follows that Xis atriodic. Thus Xis a generalized arc. Corollary 3.16. Let X={Xn, n,p mn,N}be an approximate inverse sequence of arcs and monotone onto mappings. Then X= lim X is an arc. Proof: Now, from 3.15, it follows that Xis a generalized arc. Moreover, Xis a metrizable generalized arc. Thus, Xis an arc. Acknowledgement. The author is grateful to the referee for his help and valuable suggestions. References 1. Charatonik J. J., Two invariants under continuity and the indecomposability of fans, Fund. Math. 53 (1964), 187–204. 232 I. Lonˇ car 2. Dugundji J.,“Topology,” Wm. C. Brown Publishers, Iowa, 1989. 3. Engelking R.,“General Topology,” PWN, Warszawa, 1977. 4. Freudenthal H., Entwicklungen von R¨aumen und ihre Gruppen, Compositio Math. 4(1937), 145–234. 5. Gordh G. R., On decompositions of smooth continua, Fund. Math. 75 (1972), 51–60. 6. Hall D. W. and Spencer G. L.,“Elementary Topology,” J. Wiley, New York, 1955. 7. Kuratowski K.,“Topologija I,” Mir, Moskva, 1966. 8. Kuratowski K.,“Topologija II,” Mir, Moskva, 1969. 9. Maˇ ckowiak T., On smooth continua, Fund. Math. 85 (1974), 79–95. 10. Mardeˇ si´ cS., On inverse limits of compact spaces, Glasnik math., fiz. i astr. 13 (1958), 249–255. 11. Mardeˇ si´ c S. and Rubin L. R., Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138(2) (1989), 129–144. 12. Mardeˇ si´ c S. and Watanabe T., Approximate resolutions of spaces and mappings, Glasnik Mat. 24(3) (1989), 587–637. 13. Mrowka S., On the convergence of nets and sets, Fund. math. 45(2) (1958), 237–246. 14. Nadler, S. B., Multicoherence techniques applied to inverse limits, Transactions Amer. Math. Soc. 157 (1971), 227–234. 15. Rakowski Z. M., Monotone decompositions of hereditarily smooth continua, Fund. math. 94 (1981), 119–125. 16. Ward L. E., Mobs, trees and fixed points, Proc. Amer. Math. Soc. 8(1957), 798–804. 17. Whyburn G. T., Analytic topology, Amer. Math. Soc. 28 (1971). Keywords. Approximate inverse system and limit, Freudenthal space 1991 Mathematics subject classifications: 54B25, 54D30 Fakultet organizacije i informatike Pavlinska 2 42000 Varaˇzdin CROATIA Primera versi´o rebuda el 26 de Maig de 1993, darrera versi´o rebuda el 20 de Juliol de 1995