TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 301. Definition. Let m be a positive integer, and let X be a set. An m-tuple of elements of X is a function x : f1; : : : ; mg ! X: We sometimes use xi instead of x(i); and sometimes denote x by (x1; : : : ; xm): 302. Definition. Let m be a positive integer, let fA1;:::;Amg be an indexed collection of sets, and let X = A1 [···[ Am: We define the Cartesian product of the indexed family to be the set of all m-tuples (x1; : : : ; xm) such that xi 2 Ai for Qm each i: We denote the Cartesian product by i=1 Ai or A1 × · · · × Am: If all of the m sets Ai are the same, we denote the Cartesian product by X : 303. Definition. Let X be a set. An !-tuple of elements of X is a function x : Z+ ! X: We also call such a function a sequence or an infinite sequence of elements of X: We sometimes use xi instead of x(i); and sometimes denote x by (x1; x2;::: ) or (xi)i2Z+ : 304. Definition. Let fA1;A2;::: g be an indexed collection of sets, and let X = (A1 [ A2 [ ::: ): We define the Cartesian product of the indexed family to be the set of all !-tuples (x1; x2;::: ) such that xi 2 Ai for each i: We denote the Q Cartesian product by Ai or (A1 × A2 × ::: ): In the special case where all of i2Z+ ! the sets Ai are the same, we denote the Cartesian product by X : See chapter 1, section 5 of the text for details. 305. Definition. A set S is countably infinite if and only if there is a bijection f : S ! Z+: A set is countable if and only if it is either finite or countably infinite. A set is uncountable if it is not countable. 306. Theorem. Let S be a nonempty set. The following are equivalent. 1. S is countable. 2. There is a surjective function f : Z+ :! S: 3. There is an injective function g : S ! Z+: 307. Theorem. An infinite subset of Z+ is countably infinite. 308. Corollary. Any subset of a countable set is countable. 1 2 TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 309. Theorem. A countable union of countable sets is countable. A finite product of countable sets is countable. 310. Theorem. Let X = f1; 2g: Then X! is uncountable. 311. Theorem. Let S be a set, and let P (S) denote the set of subsets of S: There is no surjective map f : S ! P (S): 312. Corollary. Let S be a set, and let P (S) denote the set of subsets of S: There is no injective map f : P (S) ! S: 313. Proposition. Let m be a positive integer, and let fX1;:::;Xmg be an Qm indexed collection of topological spaces. Let E = i=1 Xi: Let B denote the col- Qm lection of all subsets B of E such that B = i=1 Wi; where Wi is an open subset of Ei for each i: Then B is a basis for a topology on E: 314. Definition. The topology generated by the basis in the previous Proposition is called the product topology. When we consider a finite product of topological spaces, unless otherwise specified we assume that the topology on the product is the product topology. 315. Proposition. Let m ≥ 3 be a positive integer, and let fX1;:::;Xmg be Qm an indexed collection of topological spaces. Then i=1 Xi is homeomorphic to Qm−1 ( i=1 Xi) × Xm: 316. Corollary. A finite product of connected spaces is connected. 317. Corollary. A finite product of compact spaces is compact. 318. Definitions. (a) A topological space X is said to be second countable if and only if there is a countable basis for the topology. (b) A topological space X is said to be Lindelof if and only if every open cover of X has a countable subcover. (c) A subset B of a topological space X is said to be dense in X if and only if B = X: (d) A topological space X is said to be separable, if and only if there exist a countable, dense subset of X: 319. Proposition. Let X be a second countable topological space. Then X is separable. 320. Proposition. Let X be a second countable topological space. Then X is Lindelof. TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 3 321. Example. Give an example of a topological space which is Lindelof, but not separable. 322. Example. Give an example of a topological space which is separable, but not Lindelof. 323. Definitions. (a) Let I be any set, and let fXi : i 2 Ig be an indexed family of sets. The Cartesian Product of these sets is defined to be Y [ Xi = fx : I ! ( Xi): 8i 2 I; x(i) 2 Xig i2I i2I We sometimes denote an element of the Cartesian Product by (xi)i2I : (b) Let a 2 I: The projection Y pa : Xi ! Xa i2I is defined by pa(x) = x(a): 324. Axiom of Choice. The Cartesian Product of a non-empty family of non- empty sets is non-empty. 325. Definition. Let I be any set, and let f(Xi; Ti): i 2 Ig be an indexed family of topological spaces. Let Y X = Xi: i2I Let S be the collection of subsets of X of the form Y W = Wi i2I where Wi 2 Ti for each i 2 I and Wi = Xi for all except possible one i 2 I: Observe that S is a collection of subsets of X whose union is X: Let T be the topology generated by the subbasis S: Then T is called the product topology on X: Whenever we talk about the product of topological spaces, we will assume that we are using the product topology. 326. Theorem. The product of Hausdorff spaces is a Hausdorff space. Note: Here the index set I is arbitrary. 4 TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 327. Proposition. Let X and Y be topological spaces, and let f : X ! Y: Sup- pose that S is a subbasis which generates the topology on Y: Then f is continuous if and only if for each W 2 S; the inverse image f −1(W ) is an open subset of X: 328. Theorem. Suppose that Y X = Xi: i2I Suppose that f : Y ! X. Then f is continuous if and only if each composition pa ◦ f where a 2 I is continuous. 329. Proposition. Suppose that fXi : i 2 Ig and fYj : j 2 Jg are indexed families of topological spaces. Suppose that α : I ! J is a bijection and for all Q Q i 2 I; we have Xi = Yα(i): Then i2I Xi is homeomorphic to j2J Yj: 330. Proposition. Suppose that fXi : i 2 Ig and fYi : i 2 Ig are indexed families of topological spaces. Suppose that for each i 2 I; the spaces Xi and Yi Q Q are homeomorphic. Then i2I Xi and i2I Yi are homeomorphic. 331. Theorem. The countable product of metrizable spaces is a metrizable space. 1 2 332. Problem. Let A1 = [0; 1];A2 = [0; 3 ] [ [ 3 ; 1] and define An inductively by deleting the open middle third interval of each closed interval from the previous 1 stage. Let C = \n=1An: Then C is called the Cantor set. Prove that 1. C is closed (as a subset of R). 2. No open interval is a subset of C: 3. For each x 2 C we have x 2 C − fxg; so x is a limit point of C: 333. Problem. Let X denote the set f1; 2g with the discrete topology. Prove that X! is homeomorphic to the Cantor set. In particular, note that X! does not have the discrete topology. 334. Lemma. Let I be an index set, and suppose that I is the union of two disjoint subsets J and K: Let f(Xi; Ti): i 2 Ig be an indexed family of topological spaces. Then Y Xi i2I is homeomorphic to Y Y Xi × Xi i2J i2K 335. Problem. Prove the the product of the Cantor set with itself is homeomor- phic to the Cantor set. TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 5 336. Theorem. The product of connected spaces is connected. Hint: Recall that the finite product of connected spaces is connected. In the general case, consider the component of a point. 337. Definitions. A partial order on a set X is a reflexive, antisymmetric, transitive relation on X: A partially ordered set is an ordered pair (X; ≤) such that X is a set, and ≤ is a partial order on X: Let (X; ≤) be a partially ordered set, and let A ⊂ X: (a) An element w 2 X is called a maximal element of X if and only if for every x 2 X; w ≤ x implies that w = x: (b) An element u 2 X is called an upper bound of A if and only if x ≤ u for every x 2 A: (c) A is said to be a chain in X if and only if for every pair x; y 2 A either x ≤ y or y ≤ x: 338. Theorem. (Zorn's Lemma) If (X; ≤) is a partially ordered set such that every chain in X has an upper bound, then there is a maximal element of X: Note: This can be proved using the Axiom of Choice. We will not prove this here. 339. Definitions.