Immerse Metric Space Homework

Transcription

1 Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps (a) through (d) show that d(x, y) is a metric. (a) We want to show that the distance between any two points is greater than or equal to zero. We know that each x i y i is greater than or equal to 0 by the definition of absolute value. So the sum d(x, y) = x y x n y n is greater than or equal to 0. (b) Next we want to show that d(x, y) = 0 x = y. Suppose that d(x, y) = x y x n y n = 0; then each x i y i = 0 for all i =... n. So x i y i = 0 and x i = y i for all i =... n. Suppose that y = x. Then for d(x, y) = x y x n y n we can plug x i in for y i, and we get d(x, y) = x x x n x n = 0. (c) Now we want to show that d(x, y) = d(y, x). Assume d(x, y) d(y, x). Let m i = x i y i for all i =,..., n. Then x y x n + y n = y x y n x n, implying that m m n = m m n. We have reached a contradiction, since by definition m = m. Therefore, d(x, y) = d(y, x). (d) Finally we want to show that the triangle inequality holds for d. We know that x i y i = x i z i + z i y i x i z i + z i y i. Since this holds for all i =,..., n we know that d(x, y) = n x i y i n x i z i + z i y i = d(x, z) + d(z, y). Thus the triangle inequality holds for d. Now to show that d(x, y) induces the usual topology, let δ be the usual metric, let ε B δ (x, ε) be given, and let y be in the open ball centered at x of radius n, that is ε y B d (x, n ). Then we know that x y x n + y n < ε n, which implies that each individual x i y i < ε n for all i =,..., n. Therefore: n (x i y i ) 2 < n ε 2 n = ε Hence,y B δ (x, ε), which means that y is in an open ball in the topology induced by d(x, y), implies that y is in an open ball in usual topology. Now if we let B d (x, ε) be given, and let y B δ (x, ( ε n )2 ), which implies that (x y ) (x n y n ) 2 < ( ε n )2. Then we know that each x i y i 2 < ( ε n )2 for all i =,..., n, implying that x i y i < ε n for all i =,..., n. Therefore: x i y i < ε

2 Thus, y B d (x, ε). Thus, if y is in an open ball in the usual topology, then y is in an open ball in the topology induced by d(x, y). 2. In R n, define d(x, y) = ( x i y i p ) /p. Assume that d is a metric, and show that d induces the usual topology. We want to show that τ d τ Euc where τ d is the topology induced by the metric d and τ Euc is the topology induced by the usual Euclidean metric. We know by exercise () that τ Euc τ abs where τ abs is the topology induced by the absolute value metric as defined in exercise (). So we will prove that τ d τ abs. Let y B d (x, ɛ). Then ( x i y i p ) /p < ɛ. So x i y i p < ɛ p. So x i y i p < ɛ p i =,..., n. Then x i y i < ɛ i =,..., n. So x i y i < nɛ. So y B abs (x, nɛ). Now let z B abs (x, ɛ). So x i z i < ɛ. Then ( x i z i ) p < ɛ p. We know x i z i p ( x i z i ) p. So ( x i z i p ) /p z B d (x, ɛ). Hence τ d τ abs τ Euc as desired. x i z i < ɛ. So 3. Show that the topology induced by a metric d is the coarsest topology relative to which the metric is continuous Proof. ( First, show If d is continuous with metric τ, then τ d τ.) Assume that d is continuous with respect to τ. Let B be an open set in τ d such that B = B(x 0, ɛ) for some x 0 X. Show B is open in (X, τ). d : X X R + is continuous wrt τ. Define d : X R + to be d (x) = d(x, x 0 ). So d = d (X x0 ), so d is continuous wrt to τ. Note that [0, ɛ) is open in R +. Then B = B(x 0, ɛ) = d ([0, ɛ]) (X, τ). ( Next show τ d makes d continuous.) Let (a, b) R +. Show d ((a, b)) is open in τ d. Let U = d ((a, b)). 2

3 Let x d ((a, b)), so x is a point (x, y). d(x, y) (a, b). Let ɛ = min{ a d(x, y), b d(x, y) }. 2 Let B = B(x, ɛ) Xand B 2 = B(y, ɛ) X. Show B B 2 U. Let z B B 2, where z = (x, y ). Show a < d(x, y ) < b. d(x, y ) d(x, x) + d(x, y) + d(y, y ). Since d(x, x) < ɛ ( b d(x, y) ) 2 and d(y, y ) < ɛ ( b d(x, y) ), 2 d(x, y ) < b. d(x, y ) d(x, y) (d(x, x) + d(y, y)) > a, by the inverse triangle-inequality. Thus z U, so U = d ((a, b)) is open. Therefore, d is continuous on τ d. 4. Let d be a metric and let d d(x, y) (x, y) = + d(x, y). Prove that d is a bounded metric. α Claim: If α β and α, β > 0, then + α β + β. Proof. (Proof of claim) Let α β and α, β > 0. Adding αβ to both sides we obtain, α + αβ β + αβ. Now factoring out both sides, we get α( + β) β( + α). Finally, α since α, β > 0, we can divide both sides and we are left with, + α β + β. Proof. (Proof that d is a metric) Let p, q, r X.. d d(p, q) (p, q) = 0 since d(p, q) 0 and + d(p, q) > 0. Therefore, + d(p, q) d (p, q) We know that d d(p, q) (p, q) = = 0 if and only if d(p, q) = 0. Since d is a + d(p, q) metric, d(p, q) = 0 if and only if p = q. Therefore, d (p, q) = 0 if and only if p = q. 3. Since d is a metric, d(p, q) = d(q, p). So, d d(p, q) d(q, p) (p, q) = = + d(p, q) + d(q, p) = d (q, p) Therefore, d (p, q) = d (q, p). 4. Since d is a metric, d(p, q) d(p, r) + d(r, q). So by the above Lemma, d d(p, q) d(p, r) + d(r, q) (p, q) = + d(p, q) + d(p, r) + d(r, q) = d(p, r) + d(p, r) + d(r, q) + d(r, q) + d(p, r) + d(r, q). Since d is a metric, d(p, r), d(r, q) 0. Therefore d(r, q) + d(p, r) + d(r, q) d (r, q). d(p, r) + d(p, r) + d(r, q) d (p, r) and 3

4 Thus, d d(p, r) (p, q) + d(p, r) + d(r, q) + d(r, q) + d(p, r) + d(r, q) d (p, r) + d (r, q). Therefore, d (p, q) d (p, r) + d (r, q). Thus d is a metric. Proof. (Proof that d is bounded) Let p, q X. Let M =. Since d is a metric, d(p, q) 0. Therefore, d(p, q) 0, + d(p, q) > 0 and d(p, q) < + d(p, q). Hence, d d(p, q) (p, q) = + d(p, q) < = M. Therefore, d is bounded by M. 5. Let d be a metric. Show that d(x, y) = min{d(x, y), } induces the same topology as d. Proof. Let τ d and τ d be the topologies induced by d and d respectively. To show that d induces the same topology as d, we will show containment of open sets in τ d in τ d, and vice-versa. Let U τ d and fix ρ U. Then there exists r > 0 such that B d (ρ, r) U, by definition of an open set. Let r = min{r, }. Then B d(ρ, r ) = B d (ρ, r ) B d (ρ, r) U. So U τ d. Now let U τ d. Fix ρ U. Then there exists r > 0 such that B d(ρ, r ) U. We may suppose that r (else B d(ρ, r ) = X U, which implies that U = X, open in τ d ). So B d (ρ, r ) = B d(ρ, r ) U and U τ d. We have shown double containment of open sets in τ d and τ d, therefore we have that d(x, y) = min{d(x, y), } induces the same topology as d. 6. For x and y in R n, let x y = x i y i and x = x x. Show that the Euclidean metric d on R n is a metric by completing the following:. (a) Show that (x y) + (x z) = x (y + z). PROOF. (x y) + (x z) = x i y i + x i z i = (x i y i + x i z i ) = [x i (y i + z i )] = x (y + z) (b) We need to show x y x y x, y R n 4

5 Proof. We note that 0 x t y 2 = x t y x t y = x x t y t y x t y = x x t x y t y y + t 2 y y = x 2 2t x y + t 2 y 2 If y = 0, we have shown the inequality. Assume that y 0. Let t = x y y 2. Then we have from above Then, we have x ( y x 0 x 2 2 x ) 2 y y + y 2 y 2 y 2 x 2 2 ( x y ) 2 + ( x y ) 2 y 2 y 2 = x 2 ( x y ) 2 y 2 ( x y ) 2 x 2 y 2. Thus taking the square root of both sides we have, ( x y ) x y (c) Show that x + y x + y. PROOF. x + y = (x + y) (x + y) x + y 2 = (x + y) (x + y) 2 = (x i + y i ) 2 = = (x 2 i + 2x i y i + yi 2 ) x 2 i + = x x i y i + x i y i + y 2 x x y + y 2 x x y + y 2 by part (c) = ( x + y ) 2 y 2 i 5

6 Taking the squareroot of both sides we achieve the desired result. (d) Verify that d(x, y) = x y is a metric. PROOF. To verify that d is a metric it must satisfy the following four properties. i. d(x, y) 0 Proof. d(x, y) = x y = n (x i y i ) 2 0 ii. d(x, y) = 0 x = y Proof. d(x, y) = 0 x y = 0 (x y) (x y) = 0 n (x i y i ) 2 = 0 x i = y i i iii. d(x, y) = d(y, x) Proof. d(x, y) = x y = n (x i y i ) 2 = n (y i x i ) 2 = y x = d(y, x) 6

7 iv. d(x, y) d(x, r) + d(r, y) Proof. d(x, y) = x y = (x r) + (r y) x r + r y (by part (c)) = d(x, r) + d(r, y) By verifying these properties we have proven that the Euclidean metric d on R n is a metric. 7. Prove the continuity of the algebraic operations of the real line. Lemma: Let f, g : X Y be continuous functions. Then the function h : X X Y Y defined by h(x, y) = (f(x), g(y)) is continuous. From the lecture notes theorem part (i) it is sufficient to prove that the function F : X X Y defined by F (x, y) = f(x) is continuous (and similarly the function G : X X Y defined by G(x, y) = g(y) is continuous). Let U be an open set in Y. Then f (U) is open in X. But F (U) = f (U) X. Since f (U) and X are both open in X, f (U) X is open in X X, so F is continuous. Proof. (a) Addition: Let (a, b) be a basis element of R. Let f : R R R be the addition function. Then f [(a, b)] = {(x, y) R R x + y (a, b)}. Fix any (c, d) f [(a, b)] and let ɛ = min{(c + d a), (b (c + d))}. We want to show that there is an open set containing (c, d) that is contained within f [(a, b)]. Let d 2 be the taxicab metric, which we know induces the usual topology on R R. Then B d 2 ((c, d), ɛ/2) is an open set that that contains the point (c, d). Let (w, z) B d 2 ((c, d), ɛ/2). Then d 2 ((w, z), (c, d)) = w c + z d < ɛ/2, which implies that both w c and z d are less than ɛ/2. So w + z < ɛ + c + d b (c + d) + c + d = b and w + z > c + d ɛ c + d (c + d a) = a. So w + z (a, b), which implies that B d 2 ((c, d), ɛ/2) f [(a, b)]. Therefore f [(a, b)] is open, which implies that f is continuous. (b) Subtraction: It is enough to show that the additive inverse map g : R R defined by g(x) = x is continuous. Then the subtraction map s : R R R can be written as s(x, y) = f(i(x), g(y)) where i(x) is the identity map (continuous by lecture theorem part (b)) and f(x, y) is the addition map which we have shown to be continuous. Then by our lemma s is the composition of continuous functions is itself continuous. Fix any x R and any ɛ > 0. If x y < ɛ, then g(x) g(y) = x ( y) = ( )(x y) = x y < ɛ. So by our ɛ δ definition of continuity g is continuous. 7

8 (c) Multiplication Fix any ɛ > 0 and any point (x, y) R R. Then set δ = m+ m 2 +4ɛ 2 > 0, where m = max{ x, y }. Consider any (x n, y n ) B d 2 ((x, y), δ). Then x x n + y y n < δ x x n x x n y y n + y y n 2 < δ 2 x x n y y n < δ 2. From our definition of m we know We note that δ 2 + mδ = y x x n + x y y n m( x x n + y y n ) < mδ. ( m 2 4 m ) ( ) m2 + 4ɛ + m2 + 4ɛ m m m 2 + 4ɛ = ɛ Combining these results and using the triangle inequality we have y x x n + x y y n + x x n y y n < mδ + δ 2 = ɛ yx yx n + y y n ( x 0 + x x n ) < ɛ yx yx n + y y n ( x n 0 ) < ɛ yx yx n + yx n y n x n < ɛ yx y n x n < ɛ. Therefore, by our ɛ δ definition of continuity multiplication of real numbers is continuous. (d) Division It is enough to show that the multiplicative inverse map r : R/{0} R defined by r(x) = /x is continuous. Then the division map v : R R R can be written as v(x, y) = m(i(x), r(y)) where i(x) is the identity map (continuous by lecture theorem part (b)) and m(x, y) is the multiplication map which we have shown to be continuous. Then by our lemma v is the composition of continuous functions is itself continuous. Fix any ɛ > 0 and any x 0. Let δ = ɛ x 2 > 0. Consider any x +ɛ x n B(x, δ). Then x x n < δ = ɛ x 2 + ɛ x ( + ɛ x ) x x n < ɛ x 2 x x n + ɛ x x x n < ɛ x 2 x x n < ɛ x ( x x x n ). 8

9 Applying the triangle inequality we have x x n < ɛ x ( x x n + x n 0 x x n ) x x n < ɛ x ( x n ) x x n < ɛ x x n x n x < ɛ. Therefore, by our ɛ δ definition of continuity taking reciprocals of real numbers is continuous. 8. Given f, g : X R are continuous, prove that f + g, fg, f g, and f/g (provided g is nowhere zero) are all continuous. Notice that these are all functions of the form h(f(x), g(x)). For example, given h(a, b) = a + b, then f(x) + g(x) = h(f(x), g(x)). If we can show that, given continuous h : R 2 R, that h(f(x), g(x)) is continuous, then we would be done. Let p : X R 2 be defined as p(x) = (f(x), g(x)). Then we have that h(f(x), g(x)) is just h p(x). Since we know compositions of continuous functions are continuous, we just need to show that p is continuous. (See the lemma in problem 7 for the proof of this claim.) 9. In R n (and metric spaces, in general), x n x means that given ɛ > 0 there is a finite integer N such that d(x n, x) < ɛ for all n > N. Show that this agrees with the definition of convergence given for topological spaces. Problem rephrased: In topology: x n x U neighborhood of x N N such that n > N x n U In R d : x n x ɛ > 0 N N such that n > N d(x n, x) < ɛ Show these statements are equivalent. Proof: ( ) Let x n x in R d. Let U be a neighborhood of x. Then ɛ > 0 such that B(x, ɛ) U So N N such that for n > N, d(x n, x) < ɛ which implies x n B(x, ɛ) x n U. ( ) Let x n x (in topology). Given ɛ > 0, B(x, ɛ) (which is a neighborhood of x) N N such that for n > N, x n B(x, ɛ) which implies d(x n, x) < ɛ. 0. (The solution to 0 was not typed up due to a miscommunication). (The solution to was not typed up, because of its similarity to 7) 9

10 2. Using the closed set formulation of continuity show that the sets {(x, y) xy = }, {(x, y) x 2 + y 2 = }, and {(x, y) x 2 + y 2 } are closed in R 2. Proof. Recall that for f : X Y a continuous function, f (C) is closed in X whenever C is closed in Y. Let f : R 2 R be defined by f(x, y) = xy. By exercise 7, f is continuous. {} is a closed set in R. Then f ({}) = {(x, y) xy = } is closed in R 2. Similarly, let g : R 2 R be defined by g(x, y) = x 2 + y 2. Then g ({}) = {(x, y) x 2 + y 2 = } is closed in R 2. Since (, ] is a closed set, g ((, ]) = {(x, y) x 2 + y 2 } is closed in R Let f n : R R be defined by f n (x) = and let f(x) = 0. Show that n 3 (x (/n)) 2 + f n (x) f(x) for each x, but f n doesn t converge uniformly to f. Pointwise Convergence: Let ɛ > 0 and x R be given. We want to show that f n (x) converges pointwise to zero. Note that if x = 0, then f n (0) =, which clearly converges to 0 as n. When x 0, I claim that if N = Proof of claim: ɛ n+ x, then for all n > N, f n (x) < ɛ. n > ɛ x (nx ) 2 > /ɛ n 2 x 2 2nx + > /ɛ n 3 x 2 2n 2 x + n > /ɛ (since n ) n 3 (x 2 (2x/n) + (/n 2 )) > /ɛ n 3 (x (/n)) 2 > /ɛ n 3 (x (/n)) 2 + < ɛ f n (x) < ɛ. Thus, we can make f n (x) < ɛ, so we have pointwise convergence to 0. Uniform Convergence: We see that f n (/n) = for all n, so for 0 < ɛ <, there s no N such that for all x R and all n > N, f n (x) < ɛ. Thus, f n doesn t converge uniformly to (a) If {s n } is a bounded sequence of real numbers and s n s n+ for each n, then {s n } converges. Let T be the least upper bound of {s n }. Since {s n } is a bounded sequence, we know T <. I claim that the s n converge to T. Take any ball B of radius ɛ centered around T. B must contain some s N for some N, because otherwise T ɛ would be upper bound 0

11 of the s n, contradicting the leastness of T. Since it contains s N, it also contains s n for all n N, since s n are monotonically increasing. Therefore, s n converge to T. (b) Let {a n } be a sequence of real numbers. Define s n = n a i. If s n s, we say the infinite series a i converges to s. Show that if a i converges to s and b i converges to t, then ca i + b i converges to cs + t. Take some ɛ > 0. Note that if c = 0, then the result is obvious. Otherwise, if a i s, then there exists an N a such that for n N a, n a i s < ɛ. Likewise, there exists 2 c an N b such that for n N b, n b i t < ɛ/2. Let N = max{n a, N b }. Therefore, we see that for n N, ( ) ( (ca i + b i ) (cs + t) = c a i s + b i t) ( ) ɛ < c + ɛ 2 c 2 = ɛ Therefore, ca i + b i converges to cs + t. (c) (Comparison test) If a i b i for each i and b i converges then a i converges. Let c i = { a i if a i 0 0 otherwise, and let d i = { a i if a i 0 0 otherwise. Suppose b i b. Then we see that both c i and d i are bounded by b, and are strictly increasing. Therefore, by part (a), we have both c i converges, and d i converges. Suppose c i c and d i d. I claim that a i c d. Take some ɛ > 0. There exists an N c such that for n N c, n c i c < ɛ/2. Likewise, there exists an N d such that for n N d, n d i d < ɛ/2. Let N = max{n a, N b }. Then, we see that for n N, ( ) ( a i (c d) = c i c + d i d). for γ N c and δ N d. Therefore, we have a i (c d) Therefore, the a i converge. < ɛ 2 + ɛ 2 = ɛ (d) (Weierstrass M-test) Given f n : X R, and let s n (x) = n f i(x). If f i (x) b i, for all x and i, where b i converges, then s n (x) converges uniformly to a function s(x).

12 Let s(x) = f i(x) pointwise, which we know converges for each individual x because of part (c). We want to show this convergence is uniform. Suppose it were not. Then there exists an ɛ > 0 such that for any n, there exists an x such that f i (x) s(x) ɛ. We know that since b i converges, it is Cauchy. So, there exists an N such that for n, k N, k b i < ɛ for the same ɛ as above. i=n For this value of N, there exists an x 0 such that N f i (x 0 ) s(x 0 ) ɛ. We know that n f i(x 0 ) s(x 0 ). Therefore, there exists a K > N, such that for k K, we have k s(x 0) f i (x 0 ) < ɛ/2 Therefore, combining the last two inequalities with the triangle inequality, we get N f i (x 0 ) s(x 0 ) k s(x 0) f i (x 0 ) ɛ ɛ/2 N k f i (x 0 ) s(x 0 ) + s(x 0 ) f i (x 0 ) ɛ/2 k f i (x 0 ) ɛ/2 i=n+ However, we see that (using the triangle inequality again) k k f i (x 0 ) f i (x 0 ) i=n+ i=n+ k i=n+ < ɛ/2 This is a contradiction. Therefore, the convergence is in fact uniform. b i 2

13 5. Let f be a uniformly continuous real-valued function on a bounded subset E of the real line. Show that f is bounded on E. Proof. Since E is bounded, E [ M, M] for some M 0. Let ε =. Since f is uniformly continuous, choose δ > 0 such that f(x) f(y) < ε = whenever x y < δ. Let B = {B(x, δ ) x [ M, M]} Since [ M, M] is compact and B is an open cover of 2 [ M, M], there exists finite subcover B(x, δ ),..., B(x 2 n, δ ). Let x, y E B(x 2 i, δ ) 2 for some i =,, n. Then x y < δ which implies f(x) f(y) <. Hence f is bounded on each E B(x i, δ ), say by N 2 i. Since B(x, δ ),..., B(x 2 n, δ ) covers E, f is 2 bounded on E by max{n i } for i =,..., n. Show that f need not be bounded if E is not bounded. Example: Let E = R and let f(x) = x. Then f is uniformly continuous, since we are given ε > 0, and letting δ = ε, x y < δ x y < ε f(x) f(y) < ε. Finally, f is clearly not bounded on R, since given any M > 0, f(m + ) = M + > M. 0 x Q 6. Consider the function f defined by f(x) = /n x = m (m,n rel.prime, n > 0) n x = 0. Prove that f is continuous at every irrational and discontinuous at every rational. Proof. Let x 0 Q {0} such that x 0 = m 0 n 0 where m 0, n 0 are relatively prime and n 0 > 0. Fix ɛ < n 0. For any δ > 0, there is some x R Q such that x (x 0 δ, x 0 +δ). So x 0 x < δ but f(x) f(x 0 ) = 0 n 0 = n 0 > ɛ. So f is discontinuous at x 0 Q {0}. Now consider the case where x = 0. Then for ɛ < and any δ > 0, there is some x R Q such that x ( δ, δ). Again 0 x < δ but f(x) f(0) = 0 = > ɛ. We have f discontinuous at 0, so f is discontinuous for all x Q. Now let x 0 R Q and fix ɛ > 0. Let N be the smallest N N such that < ɛ. For each n < N define q N n = m such that m is the smallest integer such n that m > x n 0 and m,n relatively prime. Also define q n = m such that m is the largest n integer such that m < x n 0 and m,n relatively prime. Consider the finite set of rationals A = {q n n < N} {q n n < N} and let δ = min{d(a, x 0 ) a A}. Then x x 0 < δ means f(x) f(x 0 ) = 0 < ɛ if x R Q. And if x Q, then x = m n where m, n are relatively prime and n N, so f(x) f(x 0 ) = f(x) = < ɛ. n N 3

14 Therefore f continuous at every irrational number. 7. Let f : X (R) be a continuous function on a metric space. Show that the zero set Z f = {x f(x) = 0} is closed. The set {0} is closed in R. Therefore its continuous preimage is closed. Z f = {x f(x) = 0} 8. If A is a nonempty subset of a metric space X, define the distance from x to A to be δ A (x) = glb y A d(x, y). Prove: (a) δ A (x) = 0 x A, and (b) δ A is uniformly continuous. (a) We have that x A if and only if every open set containing x contains a point of A. But this is true if and only if every open ball about x contains a point of A. By the definition of an ɛ-ball, this is true if and only if for every ɛ > 0, there exists y A such that d(x, y) < ɛ. But this is so if and only if glb y A d(x, y) = 0, which is equivalent to δ A (x). (b) Let ɛ > 0 be given. Suppose x, x X with d(x, x ) < ɛ; we will show that δ A (x) δ A (x ) < ɛ, establishing uniform continuity. Suppose to the contrary that δ A (x) δ A (x ) > ɛ, and assume without loss of generality that δ A (x ) > δ A (x). Then we can write δ A (x ) = δ A (x) + ɛ + γ, where γ > 0. But since δ A (x) is the greatest lower bound of {d(x, z) z A}, there exists a y A such that d(x, y) < δ A (x) + γ. Then by the triangle inequality, d(x, y) d(x, x) + d(x, y) < ɛ + δ A (x) + γ = δ A (x ). But this contradicts that δ A (x ) is a lower bound for {d(x, z) z A}. 9. Let A and B be disjoint nonempty closed subsets of a metric space X. Define f : X R by for all x X. f(x) = δ A (x) δ A (x) + δ B (x) 4

15 (a) Show f is continuous and the range of f lies in [0,]. δ A δ A +δ B Note δ A and δ B are continuous on X, so δ A + δ B is as well. Therefore, f = will be continuous on X provided δ A + δ B 0 on X. But, if there exists some x X where δ A (x) + δ B (x) = 0 then δ A (x) = δ B (x) = 0 since both of the δ functions are nonnegative. Thus, by 8 a), x Ā = A and x B = B. But this means A and B are not disjoint, contradicting our assumption that they are. Therefore, δ A + δ B > 0 on X which implies f is continuous on X. To show the range property of f, note δ A 0 and δ A + δ B > 0 on X which implies f 0 on X. Also, if x X 0 f(x) = δ A (x) δ A (x) + δ B (x) δ A(x) + δ B (x) δ A (x) + δ B (x) = using the fact that δ B is a nonnegative function on X. (b) Show f(x) = 0 iff x A and f(x) = iff x B. Note f(x) = 0 iff δ A (x) = 0 iff x Ā = A. Also, f(x) = iff δ A(x) = δ A (x) + δ B (x) iff δ B (x) = 0 iff x B = B. (c) Show that every closed set A in X is the zero set for some continuous function. If A =, then choose the function identically on all of X. δ A, which vanishes identically on Ā = A. If A, then pick (d) Show that there exist disjoint open sets U and V where A U and B V. Let f be defined by f(x) = δ A (x) δ A (x) + δ B (x) on X. Let Û = [0, ) and ˆV = (, ] which are open in [0,]. Thus, U = f (Û) and 2 2 V = f ( ˆV ) are open in X by the continuity of f. Also, A = f ({0}) f ([0, )) = 2 f (Û) = U and B = f ({}) f ((, ]) = f ( ˆV ) = V. Finally, because f is a 2 function, U and V are disjoint. 20. Suppose f, g : X Y are continuous mappings between metric spaces and E is a dense subspace of X. a) Prove f(e) is dense in f(x). We must show f(e) = f(x). Since f is continuous, we know f(e) f(e). So f(e) = f(x) f(e). And f(e) f(x) implies f(e) f(x) = f(x) since f(x) is closed in f(x). So f(e) = f(x) and hence f(e) is dense in f(x). 5

16 b) Prove if f(x) = g(x) for all x in E, then f(x) = g(x) for all x in X. Let y X E. We know y E since y X. Suppose f(y) g(y). Let d = d(f(y), g(y)). So B(f(y), d) B(g(y), d) =. Let g(x 3 3 n) = f(x n ) be sequences such that x n E and d(y, x n ) <. We know these sequences exist since we know f(x) = n g(x) x E and y E. Since f is continuous, f(x n ) f(y). So there exists some N N such that f(x n ) B(f(y), d) n > N 3. Likewise, there is some N 2 N such that g(x n ) B(g(y), d) n > N 3 2. Choose m such that m > N and m > N 2. Then f(x m ) B(f(y), d) and g(x 3 m) = f(x m ) B(g(y), d). So f(x 3 m) B(f(y), d) 3 B(g(y), d ). This is a contradiction, so f(y) = g(y) must hold y X Suppose f : R 2 R 2 is one-to-one and satisfies d(x, y) = implies that d(f(x), f(y)) =. Show that d(x, y) = d(f(x), f(y) for all x, y R 2. Proof. Fix any point x R 2 and any point y 0 R 2 such that d(x, y 0 ) =. Then define points y i for i =, 2, 3, 4, 5 by rotating along the circle of radius centered at x by πi/3 radians from y 0. Since all of the triangles formed by the segments connecting the y i and x are equilateral, the y i are all separated by a distance of unit from y i+ and y i (where the subscript addition is mod 6). Since d(a, b) = implies that d(f(a), f(b)) =, the circle of radius centered at x must map into the circle of radius centered at f(x). In particular, all of the f(y i ) must lie on this circle. Once the location of f(y 0 ) is fixed on this circle, there are only two points a distance one away from f(y 0 ) that are still on the circle (each point π/3 radians away from y 0 ). Since d(y 0, y ) =, y must map to one of these two points, and this then determines where each of the remaining y i must map. Consider the perpendicular bisector of y i and y i+. There are exactly two points on the perpendicular bisector that are a distance of unit away from y i and y i+. One of these points is x, call the other point z. Then f(z) must be one of the two points that is exactly unit away from f(y i ) and f(y i+ ). One of these points is f(x), and since f is one-to-one the other point must be f(z). Continuing in this manner we see that f must map any equilateral triangular lattice of points separated by unit onto a congruent equilateral triangular lattice of points separated by unit, and that this map is determined by the image of three adjacent points (i.e. any one triangle) of the lattice. Now consider any point p on the circle of radius centered at x such that p y 0. Let θ be the angle from y 0 to p, measured in the direction of y. We want to show that f(p) must be the point on the circle of radius centered at f(x) such that the angle from f(y 0 ) to f(p) measured in the direction of f(y ) is θ. If we can show this, then we will know that d(y 0, p) = d(f(y 0 ), f(p)). Since x was chosen arbitrarily, y 0 was chosen arbitrarily on the circle of radius centered at x, and p was chosen arbitrarily among points on this circle not equal to y 0, we would know that for all points a, b such that d(a, b) < 2, d(a, b) = d(f(a), f(b)). To show that d(a, b) = d(f(a), f(b)) for the case when d(a, b) = r 2, we can repeat the above lattice creation process on a circle 6

17 of radius r/ r. Because of the arbitrary choice of x and y 0, the lattice procedure shows that d(a, b) = d(f(a), f(b)) whenever d(a, b) is an integer multiple of the original circle s radius. Before we prove our claim about f(p), note that for any two points a, b such that d(a, b) 2, we can find a point c such that d(a, c) = d(b, c) =. Then d(f(a), f(c)) = d(f(b), f(c)) =, so by the triangle inequality d(f(a), f(b)) 2. We know that f(p) must be on the circle of radius centered at f(x). Assume that the angle from f(y 0 ) to f(p) measured in the direction of f(y ) is NOT θ. We know that the equilateral triangular lattice of points separated by unit of which p and x are a part must be mapped onto a congruent equilateral triangular lattice of points separated by unit. In particular, each point on the ray xp that is a distance r Z >0 from x must be mapped to the point on the ray f(x)f(p) a distance r from f(x). So assume that the angle from f(y 0 ) to f(p) measured in the direction of f(y ) is α θ. Let β = α θ. Consider the point on 2 xp a distance r = from x. Call this sin(β/2) point a. Let f(q) be the point such that the angle from f(y 0 ) to f(q) in the direction of f(y ) is θ, and the distance from f(q) to f(x) is r. Let b be any point on the equilateral triangular lattice of points separated by unit determined by x and y 0 such that d(a, b) < 2. Then f(a) must be the point on f(x)f(p) a distance r from f(x), which means that f(a) has been rotated by an angle β about f(x) relative to the image of the equilateral triangular lattice determined by f(x) and f(y 0 ). This corresponds to a shift of distance 2r sin(β/2) 4 for the point f(a) from the point f(q), which by the triangle inequality implies that d(f(a), f(b)) > 2. But this contradicts our above claim that any pair of points less than or equal to two units apart must map to a pair of points less than or equal to two units apart. Therefore f(p) must be the point on the circle of radius centered at f(x) such that the angle from f(y 0 ) to f(p) measured in the direction of f(y ) is θ, completing our proof. 7