MH 7500 THEOREMS De¯nition. A topological space is an ordered pair (X; T ), where X is a set and T is a collection of subsets of X such that (i) ; 2 T and X 2 T ; (ii) U \ V 2 T whenever U; V 2 T ; (iii) [U 2 T whenever U ½ T . T is called a topology on the set X. Elements of T are called open sets. Remark. To put it in other words, condition (ii) says that pairwise (and hence ¯nite) intersections of open sets are open, and (iii) says that any union of open sets is open. Convention. We will often write X is a topological space or just X is a space to mean that X is a set with some ¯xed but unspeci¯ed topology. De¯nition. If X is a space, and A ½ X, then A is closed i® X n A is open. Lemma 1. If X is any set, and A any collection of subsets of X, then T (i) X n [A = SfX n A : A 2 Ag; (ii) X n \A = fX n A : A 2 Ag. Corollary 2. For any space X, (i) ; and X are closed; (ii) The union of ¯nitely many closed sets is closed; (iii) The intersection of any number of closed sets is closed. De¯nition. If X is a space, and A ½ X, then the closure of A, denoted by A, is de¯ned to be the intersection of all closed sets which contain A. Note that therefore A is always a closed set and A is closed i® A = A. If x 2 X and if every open set containing x contains a point of A other than x, we say that x is a limit point of A. Theorem 3. For any subset A of a space X, we have A = A [ fx 2 X : x is a limit point of Ag. Theorem 4. If A and B are subsets of a space X, then (i) ; = ;; (ii) A ½ A; (iii) A = A; (iv) A [ B = A [ B. Theorem 5. If fA® : ® 2 ¤g is any collection of subsets of a space X, then T T (i) A ⊆ A ; S®2¤ ® S®2¤ ® (ii) ®2¤ A® ¶ ®2¤ A®. De¯nition. If A is a subset of a space X, let Ao = fx 2 A : x 2 U ½ A for some open set Ug. We call Ao the interior of A, and if x 2 Ao we say that A 1 2 MH 7500 THEOREMS is a neighborhood of x. A point x is in the boundary of A if every open set containing x (equivalently, every neighborhood of x) meets both A and X n A. We denote the set of boundary points of A by @A. Theorem 6. For any subset A of a space X, A = A [ @A = Ao [ @A. De¯nition. Let (X; T ) be a topological space, and let B be a subcollection of T . We say B is a base for the topological space (X; T ) i® every member of T is a union of members of B. Theorem 7. Let X be a set and let B be a collection of subsets of X. Then there is a (unique) topology T on X such that B is a base for (X; T ) if and only if (a) Each x 2 X is in some member of B (i.e., B covers X); (b) Whenever B1;B2 2 B and x 2 B1 \ B2, there exists B 2 B with x 2 B ½ B1 \ B2. De¯nition. Collections B1 and B2 of subsets of a set X are equivalent bases i® there is a topology T on X such that B1 and B2 are both bases for (X; T ). Theorem 8. Let (X; T ) be a topological space, let B1 be a base for (X; T ), and let B2 be a collection of subsets of X. If (i) x 2 B1 2 B1 ) 9B2 2 B2 such that x 2 B2 ½ B1, and (ii) x 2 B2 2 B2 ) 9B1 2 B1 such that x 2 B1 ½ B2, then B1 and B2 are equivalent bases. De¯nition. A subspace of a topological space (X; T ) is a pair (A; TA), where A ½ X and TA = fU \ A : U 2 T g. TA is called the relative topology on A, or the topology on A induced by T . Theorem 9. If (X; T ) is a topological space, B a base for T , and A ½ X, then the collection fB \ A : B 2 Bg is a base for (A; TA). De¯nition. Let X be a set. A function d : X £ X ! R is called a metric on X if for each x; y; z 2 X, d satis¯es: (i) d(x; y) ¸ 0 and d(x; y) = 0 () x = y; (ii) d(x; y) = d(y; x) (i.e., d is symmetric); (iii) d(x; z) · d(x; y) + d(y; z) (the triangle inequality). A metric space is a pair (X; d), where X is a set and d is a metric on X. If ² > 0, let Bd(x; ²) = fy 2 X : d(x; y) < ²g (if it is understood what metric we are talking about, we may omit the subscript d and write B(x; ²)). Bd(x; ²) is the ²-ball about x (w.r.t. d). Lemma 10. Let (X; d) be a metric space. Then the collection B = fBd(x; ²): x 2 X; ² > 0g of ²-balls is a base for a topology on X. Lemma 10.5. Let (X; d) be a metric space. Then a subset U of X is open in the metric topology i®, for every x 2 U, there is ² > 0 such that Bd(x; ²) ½ U. De¯nition. Let (X; d) be a metric space. Then the topology on X given in Lemma 10 is called the metric topology (generated by the metric d). If d and d0 are two metrics on the same set X which generate the same topology, then d and d0 are called equivalent metrics. A topological space (X; T ) is said to be metrizable if there is a metric on X which generates T . MH 7500 THEOREMS 3 Theorem 11. Let (X; d) be a metric space, and let Y ½ X. Let dY be the metric d restricted to Y £ Y . Then (Y; dY ) is a metric space, and the metric topology on Y is the same as the subspace topology with respect to the metric topology on X. De¯nition. If f : X ! Y , where X and Y are spaces, and x0 2 X, we say f is continuous at x0 if, given any open set V in Y with f(x0) 2 V , there is an open set U in X with x0 2 U and f(U) ½ V . Also, f is continuous if f is continuous at every x0 2 X. Theorem 12. If f : X ! Y , where X and Y are spaces, then the following are equivalent: (i) f is continuous; (ii) f ¡1(V ) is open in X whenever V is open in Y ; (iii) f ¡1(C) is closed in X whenever C is closed in Y ; (iv) Whenever A ½ X and x 2 A, then f(x) 2 f(A). Theorem 13. If f : X ! Y is continuous, and A ½ X, then the restriction of f to A, denoted f ¹ A, is a continous function from A (with the subspace topology) to Y . Theorem 14. If f : X ! Y and g : Y ! Z are continuous, so is g ± f : X ! Z. De¯nition. A function f : X ! Y is closed (resp., open) if the image of every closed set (resp., open set) in X is closed (resp., open) in Y . De¯nition. A function h : X ! Y is a homeomorphism if h is one-to-one, onto, and both h and h¡1 are continuous. If such a homeomorphism exists, then the spaces X and Y are said to be homeomorphic. Theorem 15. If h : X ! Y is one-to-one and onto, then the following are equiv- alent: (i) h is a homeomorphism; (ii) h is open and continuous; (iii) h is closed and continuous. Exercise. Find an example of spaces X and Y , and a mapping f : X ! Y which is one-to-one, onto, and continuous, but not a homeomorphism. De¯nition. A local base at a point x of a space X is a collection Bx of open neighborhoods of x such that, whenever x 2 U where U is open, there is some B 2 Bx with x 2 B ½ U. A space X is ¯rst-countable if every point of x has a countable local base. Remark. If U1;U2;::: is a countable local base at x, and we put V1 = U1;V2 = U1 \ U2;V3 = U1 \ U2 \ U3;::: , then V1;V2;::: is another countable local base at x which is decreasing (in the sense that V1 ⊃ V2 ⊃ V3 ⊃ ::: ). So a point has a countable local base i® it has a countable decreasing local base. Theorem 16. Every metric space is ¯rst-countable. 1 De¯nition. Let X be a topological space, x 2 X, and (xn)n=1 a sequence 1 1 of points in X. We say (xn)n=1 converges to x, and write (xn)n=1 ! x, if every (open) neighborhood of x contains xn for su±ciently large n 2 N (i.e., for any (open) neighborhood Nx of x, there is k 2 N such that xn 2 Nx for all n ¸ k).