Topology: Product Space

TOPOLOGY CLASS 11: PRODUCT SPACES PROF. SEBASTIAN VATTAMATTAM 0.1. The Box Topology. Denition 0.1. p:113 in [2] Let J be an index set. X 6= . A J-tuple of elements in X is a function x : J ! X: XJ := fxjx : J ! Xg If 2 J; x:= x(): x is represented by (x)2J Denition 0.2. Let (A)2J be an indexed family of sets. X :=[fAj2 Jg The cartesian product of this indexed family is dened as fAj2 Jg = f(x)2J jx2 Ag If A= X; 8then A= XJ Theorem 0.3. Let (X; )2J be an indexed family of topological spaces. If U2 ; consider the cartesian product fUj2 Jg The collection of all such sets is a basis for a topology on fXj2 Jg Proof Let X := Xand B be the collection of sets of the form U; U2 12 PROF. SEBASTIAN VATTAMATTAM (1) Let (x) 2 X ) (x) 2 fXj2 Jg 2 B since X2 (2) Let B= U;B2 = V;B;B2 2 B B1\B2 = (U\V) Since U\V2 ;B1\B2 2 B If x 2 B1TB2 then x 2 B3 =B1\B2 Denition 0.4. The topology generated by the cartesian products of open sets in Xis called the box topology. 1. The Product Topology Denition 1.1. Projection Mapping Let (X; ) be an indexed family of topological spaces. For = dene : X! Xby ((x)) = xare called projections. Example 1.2. Let J = f1; 2; 3g and (X; )2J be an in- dexed family of topological spaces. For 2 J; dene : X1 X2 X3 ! Xby ((x1; x2; x3)) = xIf x = (x1; x2; x3); 1(x) = x1; 2(x) = x2; 3(x) = x3 are the three projections. Denition 1.3. For 2 J; let U2 1 (U) = fxj(x) 2 Ug = fxjx2 UgTOPOLOGY 3 For 2 J; let S:= f1 (U)jU2 g and S := [2J SThe topology generated by S as a sub-basis is called the product topology. Xwith this topology is called the product space. Example 1.4. Let J = f1; 2g 1 (U) = fx = (x1; x2)j(x) 2 Ug = fxjx2 Ug 1 1 (U1) = f(x1; x2)jx1 2 U1g = U1 X2 , a vertical strip. 1 2 (U2) = f(x1; x2)jx2 2 U2g = X1 U2 , a horizontal strip. S1 := f1 1 (U1)jU1 2 1g = fU1 X2jU1 2 1g S2 := f1 2 (U2)jU2 2 2g = fX1 U2jU2 2 2g and S = S1[S2 The topology generated by S as a sub-basis is called the product topology. X1 X2 with this topology is called the product space. 2. Box Topology and Product Topology Let (Xi; i); i 2 f1; 2; 3g be the topological spaces. Let B be the basis for the product topology. That is B is the collection of nite intersections of elements of S Let B1 = 1 i (Ui);B2 = 1 i (Vi); for some i = 1; 2; 34 PROF. SEBASTIAN VATTAMATTAM ThenB1\B2 = 1 i (Ui\Vi) 2 Si since Ui\Vi 2 i That is each Sis closed under nite intersection. Let 1 = 1; 2 = 3 and U1 = U1 2 1; U2 = U3 2 3 B := 1 1 (U1)\1 2 (U2) = 1 1 (U1)\1 2 (U2) = (U1 X2 X3)\(X1 X2 U3) = U1 X2 U3 = fUjU= X; if =2 f1; 2gg = U1\U2\U3; U2 = X2 Theorem 2.1. 19:1 in [2] (X) a family of topological spaces, Bb is the basis for the box topology and Bp the basis for the product topology for X. Then (1) Bb = fUjU2 g (2)Bp = fUjU2 ; U= X; for almost all g References [1] W. Beekmann,Analysis 1,Fern University in Hagen [2] James R. Munkres,Topology, Second Edition, Prentice-Hall of India, New Delhi, 2002. [3] G. F. Simmo0ns, Introduction to Topology and Modern Analysis,International Student Edition, McGraw-Hill Kogakush Ltd, 1963.