MTH 304: Metric Spaces and Topology Semester 2, 2016-17 Contents 1 Topological spaces2 1.1 Basic definitions and examples..................2 1.2 Basis and subbasis for a topology................3 1.3 The Product Topology X × Y ..................4 1.4 Subspace topology........................5 1.5 Closed sets and limit points...................6 1.6 T1 and Hausdorff (T2) spaces...................8 1.7 Continuous functions.......................9 1.8 Product and box topologies for arbitrary products....... 11 1.9 Metric topology.......................... 12 1.10 Quotient topology......................... 15 1.11 Connected and path connected spaces.............. 19 1.12 Compactness........................... 23 1.13 Countability axioms....................... 26 1.14 Separation axioms........................ 29 1.15 Imbeddings of manifolds..................... 30 1 1 Topological spaces 1.1 Basic definitions and examples (i) Let X be a set, and T a collection of subsets of X such that (a) ;;X 2 T , (b) for an arbitrary index set J, and a subcollection fVαgα2J ⊂ T , we have [ Vα 2 T α2J (i.e T is closed under arbitrary union), and (c) for an finite subcollection fW1;:::Wng ⊂ T , we have n \ Wi 2 T : i=1 Then T is said to define a topology on X, and the pair (X; T ) is called a topological space. (ii) Let (X; T ) be a topological space. Then each U 2 T is called an open set. (iii) Let X be a set, and T ; T 0 be topologies on X. Then T is a said to coarser (or equivalently, T 0 is said to be finer than T ) if T ⊂ T 0. (iv) Examples of topological spaces. Let X be any set. (a) The collection f;;Xg defines a topology on X called the indiscrete topology on X. This is the coarsest possible topology that can be defined on X. (b) The power set P(X) of X defines a topology on X called the discrete topology on X. P(X) is the finest possible topology that can be defined on X. (c) The collection fS ⊂ X j jX n Sj < 1g defines topology on X called the cofinite topology on X. (d) The collection fS ⊂ X j X n S is countableg defines a topology on X called the cocountable topology on X. 2 1.2 Basis and subbasis for a topology (i) Given a set X, and a collection B of subsets of X such that (a) for each x 2 X, there exists B 2 B such that x 2 B, and (b) for any pair of elements B1;B2 2 B and y 2 B1 \ B2, there 0 0 exists B 2 B such that y 2 B ⊂ B1 \ B2. Then B is said to form a basis for a topology T on X (or B generates a topology T on X) if for each U 2 T and for each x 2 U, there exists B 2 B such that x 2 B ⊂ U. (ii) Let (X; T ) be a topological space. Let B ⊂ T be any subccol- lection such that for each U 2 T and each x 2 U, there exists B 2 B such that x 2 B 2 U. Then B generates the topology T . (iii) Let X be a set, and B a basis for a topology T on X. Then T is the collection of all arbitrary unions of elements in B. (iv) Let B and B0 be bases for topologies T and T 0 respectively, on X. Then T ⊂ T 0 if and only if for each B 2 B and each x 2 B, there exists B0 2 B0 such that x 2 B0 2 B. (v) Examples of bases for topologies on X = R. (a) The collection of all open intervals B = f(a; b) j a; b 2 R and a < bg forms a basis for a topology on R called the standard topology. Note that the real line with this topology is simply denoted by R. (b) The collection B` = f[a; b) j a; b 2 R and a < bg is a basis a for a topology R` on R called the lower limit topology. (c) The collection BK = f(a; b) j a; b 2 R and a < bg[f(a; b)nK j a; b 2 R and a < bg; 1 where K = f n j n 2 Ng; is a basis a for a topology RK on R called the K-topology. The topologies RK and R` are strictly finer than standard topology R, but they are not comparable. 3 (vi) Let X be set, and let S be any collection of subsets of X whose union equals X. Then the collection B of all finite intersections of elements in S forms a basis for a topology T on X. Any such collection S is called a subbasis for the topology T , or in other words, S generates T . 1.3 The Product Topology X × Y (i) Let (X; T X ) and (Y; T Y ) be topological spaces. Then the collection B = fU × V j U 2 T X and V 2 T Y g forms a basis for a topology on X × Y called the product topology. The product topology is simply denoted by X × Y . (ii) Let BX be a basis for (X; T X ) and By a basis for (Y; T Y ). Then the collection BX×Y = fB × C j B 2 BX and C 2 BY g forms a basis for the product topology X × Y . (iii) Examples for product topology on Rn. (a) The collection f(a; b) × (c; d)g of all open rectangles in R2 (i.e. products of open intervals in R) forms a basis for a topology on R2 called the standard topology on R2. (b) Generalizing Example (i), the collection n Y f (ai; bi)g i=1 of all open n-dimensional cubes in Rn (i.e. products of n open intervals in R) forms a basis for a topology on Rn called the standard topology on Rn (denoted simply by Rn). (c) For x 2 Rn and r > 0, consider the open ball n Bx;r = fy 2 R j ky − xk < rg: 4 Then the collection n fBx;r j x 2 R and r > 0g also forms a basis for the standard topology Rn. In the subject of Real Analysis, this is most preferred topology on Rn. (iv) For 1 ≤ i ≤ n, let (Xi; T i) be topological spaces, and let n Y pi : Xj ! Xi j=1 be the projection map onto the ith coordinate. Then the collection −1 S = fpi (Ui) j 1 ≤ i ≤ n and Ui 2 T ig n Y forms a subbasis for the product topology Xi. i=1 1.4 Subspace topology (i) Let (X; T ) be a topological space, and Y ⊂ X. The collection T Y = fU \ Y j U 2 T g forms a topology on Y called the subspace topology or the topology in- herited by Y from x. Under this topology, Y is called a subspace of X. (ii) Let (X; T ) be a topological space generated by a basis B. Then the collection BY = fB \ Y j B 2 Bg is a basis for T Y . (iii) Let Y be a subspace of (X; T ). If V 2 T y and Y 2 T , then V 2 T . (iv) Examples of subspaces. 5 (a) Let X = R and Y = [0; 1]. Then every basic open set in the subspace topology inherited by [0; 1] from R has one the following four types: 8(a; b); if a; b 2 Y; > <>[0; b); if only b 2 Y; (a; b) \ Y = >(a; 1]; if only a 2 Y; > :Y; ; if both a; b2 = Y: (b) The set Z of integers, and the set N of natural numbers both inherit the discrete topology from R. 1.5 Closed sets and limit points (i) Let (X; T ) be a topological space. Then A ⊂ X is called a closed set if X n A 2 T . (ii) Examples of closed sets. (a) In any topological space (X; T ), ; and X are closed sets. (b) The closed interval [a; b] is a closed set in R. In general, the product of n closed intervals n Y [ai; bi] i=1 is closed in Rn. (c) In the subspace Y = [0; 1] [ (2; 3) of R, the subsets [0; 1] and (2; 3) are both open and closed. (d) In the cofinite topology, all finite sets are closed. (e) All sets are closed in the discrete topology. (iii) Arbitrary intersections and finite unions of closed sets in a topological space are also closed. (iv) Let Y be a subspace of a topological space (X; T ). Then A is closed in Y if, and only if A = C \ X for some C closed in X. (v) The (X; T ) be a topological space, and A ⊂ X. 6 (a) The interior of A (denoted by A◦) is the union of all open sets contained in A, that is, A◦ = [fV 2 T j V ⊂ Ag (b) The closure of A (denoted by A¯) is the intersection of all closed sets containing A, that is, A¯ = \fC j X n A 2 T andA ⊂ C:g (c) It follows by definition that A◦ ⊂ A ⊂ A:¯ ¯ (vi) Let Y be a subspace of a topological space X, and let A ⊂ Y . Let AY denote the closure of A in Y . Then ¯ ¯ AY = A \ Y: (vii) Let (X; T ) be a topological space, and A ⊂ X. Then x 2 A¯ if, and only if, for every open set U 3 x, we have that U \ X 6= ;: (viii) Let (X; T ) be a topological space.