Cell Structures E.D. Tymchatyn (with W. Debski) University of Saskatchewan [email protected] July 26, 2013 E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 1 / 20 Theorem X compact metric space =) 9 f : C X (onto). E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 2 / 20 Proof. f0 U0 = fCg −! V0 = fX g " i0;1 " j0;1 f1 U1 = fU1;1;:::; U1;n1 g −! V1 = fV1;1;:::; V1;n1 g; mesh < 1 disjoint clopen cover closed cover " i1;2 " j1;2 f2 U2 = U2;1 [···[U2;n1 −! V2 = V2;1 [···[ V2;n1 ; mesh < 1=2 U2;i disjoint cover V2;i closed cover of V1;i of U1;i by clopen sets " i2;3 " j2;3 :::::::::::::::::: f~=(f ;f ;::: ) C ≈ lim(U ; i ) −!0 1 V = lim(V ; i ) − i j;k 1 − i j;k Define ∼ on V1 by (x0; x1;::: ) ∼ (y0; y1;::: ) iff xi \ yi 6= ? for all i π : V1 !V1= ∼ ≈ X f = π ◦ f~ E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 3 / 20 Cell structures are inverse systems of discrete spaces with combinatorial \nearness" relations. Natural quotients (induced by the nearness relations) of the inverse limits yield topological spaces. \Maps" between cell structures yield mappings between the corresponding quotient spaces. Freudenthal (1937). Every compact metric space X admits a polyhedral inverse sequence with surjective bonding maps whose inverse limit is X . E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 4 / 20 Definition Let Σ be a class of compact, connected polyhedra. Then the class [Σ] of Σ − like continua consists of all continua X such that there exist continuous mappings of X with arbitrarily small fibers onto some members of Σ. Let (Σ) be class of continua X such that X ≈ lim(P ; p 0 ) where − n α,α Pα 2 Σ and pα,α0 are continuous surjections. Note:Σ ⊂ (Σ) ⊂ [Σ]: Mardesic-Segal (1963). Let Σ be a class of compact, connected polyhedra. metric \[Σ] = metric \(Σ) eg. 1)Σ= f[0; 1]g, sin −1=x curve is arc-like i.e. sin −1=x curve 2 [Σ]. 2)Σ= fconnected, finite, polyhedra of dim ≤ ng. Then metric \[Σ] = fmetric continua of dim ≤ ng. 3) Pasynkov, Mardesic (1959) - non-metric case problems - dim 9 X non-metric, X 2 [f[0; 1]g] but X 2= (f[0; 1]g). E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 5 / 20 Mardesic (1963). Let Σ be a class of compact, connected polyhedra. [Σ] = class of lim(X ; p 0 ) − α α,α where Xα are metric Σ-like continua, dim(Xα) ≤ dim(X ) and all pα,α0 are continuous surjections. Corollary Every continuum X in [Σ] is the limit of a double iterated inverse system of polyhedra in Σ with dim ≤ dim(X ). E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 6 / 20 For mappings inverse limits are more problematic. Mioduszewski (1963). If X ; Y metric compacta and f : X = lim(P ; p ) −! Y = lim(Q ; q ) − i i;j − i i;j then for each sequence f"i g of positive numbers i ! 0 there exists p pm1;m2 pm2;m3 mk ;mk+1 Pm1 −−−− Pm2 −−−− Pm3 −−−− ::: −−−− Pmk −−−−− ::: ? ? ? ? ? ? ? ? f1y f2y f3y ::: fk y ::: q qn1;n2 qn2;n3 nk ;nk+1 Qn1 −−−− Qn2 −−−− Qn3 −−−− ::: −−−− Qnk −−−−− ::: where m1 < m2 < m3 < : : : , n1 < n2 < n3 < : : : and each diagram Pmk −−−− Pmr ? ? ? ? y y Qni −−−− Qnk −−−− Qnr is "k commutative for i < k < r. E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 7 / 20 Mardesic (1963). Let Σ and J be classes of connected polyhedra. Given X 2 [Σ] , Y 2 [J] and f : X ! Y a continuous surjection, then there exist 0 0 inverse systems X = lim(X ; p 0 ) and Y = lim(Y ; p 0 ) of metric − α α,α − β β,β continua in [Σ] (resp. [J]) with surjective bonding maps and homeomorphisms h and k so X −!f Y ≈ h # ,! ≈ k # 0 fβ 0 X = lim(X ; p 0 ) −! Y = lim(Y ; p 0 ) − α α,α − β β,β Note : Xα and Yβ can not be taken to be polyhedra even if X and Y are metric. Mardesic (1981). Resolutions - inverse systems with additional conditions to study noncompact cases. Mardesic - Watanabe (1989). Approximate inverse systems and approximate resolutions to obtain rather arbitrary spaces and mappings and filling in the deficiencies indicated above. E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 8 / 20 Definition Graph is ordered pair (G; r). G is discrete set and r is reflexive and symmetric relation on G. Cells are points of G. a; b are adjacent if (a; b) 2 r. st(a) = fb 2 G j (a; b) 2 rg. E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 9 / 20 Let f(Gi ; ri ) j i = 1; 2;::: g be graphs and gi;j : Gj ! Gi be functions for j ≥ i satisfying i) gi;i = identity on Gi ii) gi;k = gi;j ◦ gj;k for i < j < k and iii)( a; b) 2 ri+1 =) (gi;i+1(a); gi;i+1(b)) 2 ri g1;2 g2;3 g3;4 G1 −−−− G2 −−−− G3 −−−− ::: If a 2 Gi , say deg(a) = i. Let Π = ΠGi topological product. Π is complete, 0-dimensional, metric space. G = lim(G ; g ): 1 − i i;j If x = (x(1); x(2);::: ) 2 Π then x 2 G1 iff gi;i+1(x(i + 1)) = x(i) for each i. G1 is set of threads. E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 10 / 20 Let gi : G1 ! Gi be ith coordinate projection. −1 If a 2 Gi let < a >= fx 2 G1 j x(i) = ag = gi (a). f< x(i) >: x 2 G1; i = 1; 2;::: g is basis of open and closed sets for G1. Proposition G1 is closed in Π hence G1 is topologically complete 0-dimensional metric space eg. if each Gi is countable, G1 is closed subset of irrationals. Definition Set x ∼ y in G1 if (x(i); y(i)) 2 ri for each i. ∼ is reflexive and symmetric relation on G1. Proposition ∼ is closed in G1 × G1 . Proof. T ∼= Ri where Ri = f(x; y) 2 G1 × G1 j (x(i); y(i)) 2 ri g. Ri is closed since ri is closed in discrete space Gi × Gi . E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 11 / 20 Definition Cells a 2 Gm and b 2 Gn are close if (gk;m(a); gk;n(b)) 2 rk for k = minfm; ng. Cauchy Sequence is a sequence of cells fu(j)g in [Gi such that iv) lim deg(u(j)) = 1 and v) u(i) and u(j) are close for all i and j sufficiently large. Cauchy sequence fu(i)g converges to thread x 2 G1 if x(i) and u(j) are close for all i and sufficiently large j. Note. A Cauchy sequence may converge to different threads. E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 12 / 20 Definition A cell structure is an inverse sequence of graphs satisfying vi) 8 thread x 2 G , 8i, 9j ≥ i s.t. g (st2 (x(j))) ⊂ st (x(i)) 1 i;j rj rj vii) 8 thread x, 8i, 9j ≥ i s.t gi;j (strj (x(j))) is finite. viii) each Cauchy sequence of cells converges eg. Gi = f1; 2;::: g, ri = ∆ [ f(k; l) j k; l ≥ ig and gi;j = identity. f((Gi ; ri ); gi;j )g satisfies vi) and vii) but not viii). x = fx(i) = ig is Cauchy but does not converge. Note. In general if vi) and vii) are satisfied and each set of mutually adjacent cells is finite then viii) is satisfied. Let (∗) = f((Gi ; ri ); gi;j )g be a cell structure. ∼ is transitive by vi). E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 13 / 20 Definition For x 2 G1 let [x] = fy 2 G1 j x ∼ yg. By vii) [x] is compact. Define ∗ π : G1 ! G1= ∼= G : π is perfect mapping. G ∗, the space determined by the cell structure, is topologically complete metrizable space. Proposition If (∗) is cell structure then ∗ ∗ fG n π(Gi n < A >) j A ⊂ Gi ; i = 1; 2;::: g is basis for topology on G . −i eg. Gi = fp10 j p integer g −i (x; y) 2 ri iff jx − yj ≤ 10 : gi;j an order preserving retraction. G∗ ≡ R: E.D.Tymchatyn (University of Saskatchewan) Cell Structures July 26, 2013 14 / 20 Theorem (1) Each complete metric space is homeomorphic to a space determined by some cell structure. Proof. X paracompact −! 9fUi g sequence of locally finite open covers with mesh Ui < 1=i and Ui+1 closed star refines Ui . Define gi;i+1 : Ui+1 !Ui by gi;i+1(U) ⊃ cl(st(U; Ui+1)): Define ri = fU × V j U; V 2 Ui and U \ V 6= ?g. f((Ui ; ri ); gi;j )g is a cell structure. vi) follows from closed star refinement of Ui+1 in Ui . vii) is local finiteness of covers S viii) if fu(i)g is Cauchy sequence in Ui , most pairs (u(i); u(j)) intersect so form a Cauchy sequence in X converging to a point x in X . Choose inductively v(i) in Ui so x 2 v(i) and v = (v(1); v(2);::: ) ⊂ G1. Then fu(i)g converges to v.