70. Let Y be a metrizable topological space and let A Ď Y. Show that Cl Y A scl Y A.

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1 Homework for MATH 4603 (Advanced Calculus I) Fall 2017 Homework 14: Due on Tuesday 12 December 66 Let s P pr 2 q N let a b P R Define p q : R 2 Ñ R by ppx yq x qpx yq y Show: r s Ñ pa bq in R 2 s ô r ppp sq Ñ a in Rq & ppq sq Ñ b in Rq s 67 Let Y be a topological space let A Ď Y let z P Y Show: p z P Cl Y A q ñ P N Y pzq V X A H q 68 Let Y be a topological space let A Ď Y Show: scl Y A Ď Cl Y A 69 Let py dq be a metric space let s P Y N let z P Y Show: p s Ñ z in Y q ô p dps zq Ñ 0 in R q 70 Let Y be a metrizable topological space let A Ď Y Show that Cl Y A scl Y A Homework 13: Due on Tuesday 5 December 61 Define s P R N by s p 1q Show: (a) s Ñ 1 (b) s Ñ 1 62 Let f : R Ñ R 2 be defined by fptq p3t 6 4t ` 1q Show that f is uniformly continuous 63 Let f : R Ñ R be defined by fptq t 2 Show that f is NOT uniformly continuous 64 Let Y be a metric space let Z be a normed vector space Let f g : Y Z Assume that both f g are uniformly continuous Show that f ` g is uniformly continuous 65 Let Y : r1 2q Y r3 4q Z : r5 7q Define f : Y Ñ Z by # s ` 4 if s P r1 2q fpsq s ` 3 if s P r3 4q

2 Show (a) f : Y ãñą Z (b) f is continuous (c) f 1 is NOT continuous Homework 12: Due on Tuesday 28 November 56 Let f g : R R let a P R Assume: Show: fg Ñ 8 near a r p f Ñ 8 near a q p g Ñ 8 near a q s 57 Let Y be a topological space Let g : Y R let a P Y Assume: g Ñ 8 near a Show: 1{g Ñ 0 near a 58 Let Y be a topological space Let f g : Y R let a P Y Assume: DP P N ˆpaq such that f ď g on P Assume: f Ñ 8 near a Show: g Ñ 8 near a 59 Find f : R Ñ R g : R Ñ R such that f Ñ 2 near 1 g Ñ 3 near 2 g f Ñ 4 near 1 60 Let X Y Z be topological spaces Let f : X Y let g : Y Z Let a P X let b P Y Assume: f Ñ b near a Assume: g is continuous at b Show: g f Ñ gpbq near a Homework 11: Due on Tuesday 21 November 51 Let Y Z be topological spaces Y 0 Ď Y Z 0 Ď Z f : Y 0 Z 0 a P Y 0 b P Z 0 Assume that f Ñ b in Z 0 near a in Y 0 Show that f Ñ b in Z near a in Y 52 Let f g : R R a P R Assume that f Ñ 8 in R near a in R that g Ñ 8 in R near a in R Show that f ` g Ñ 8 in R near a in R 53 Find functions f g h : R Ñ R such that (1) f Ñ 8 in R near 8 in R (2) g Ñ 8 in R near 8 in R (3) h Ñ 0 in R near 8 in R

3 (4) fh Ñ 8 in R near 8 in R (5) gh Ñ 0 in R near 8 in R 54 Let s P pr q N assume that s 1 ď s 2 ď s 3 ď Define z : supts 1 s 2 s 3 u assume that z 8 Show: s Ñ z in R 55 Find s t σ τ P R N such that P N [ ( σ 0 τ ) ( σ ` τ 0 ) ] (2) ps{σq Ñ 1 in R (3) pt{τq Ñ 1 in R (4) pps ` tq{pσ ` τqq Ñ 2 in R Homework 10: Due on Tuesday 14 November 46 Let Z be a topological space let W be an open subset of Z let S Ď W Show: ( S is open in Z ) ô ( S is open in W ) Note: The topology on S is the relative topology inherited from Z 47 Let Z be a topological space let C be a closed subset of Z let S Ď C Show: ( S is closed in Z ) ô ( S is closed in C ) Note: The topology on C is the relative topology inherited from Z 48 Let Z be a topological space C : tclosed subsets of Zu Show: xcy Z X C xcy fin Y 49 Let Z be a topological space let Z 0 be an open subset of Z let p P Z 0 Let B 0 be a neighborhood base at p in Z 0 Show that B 0 is a neighborhood base at p in Z Note: The topology on Z 0 is the relative topology inherited from Z 50 Show that B 8 is a neighborhood base at 8 in R Note: The topology on R is T : xb y Y Homework 9: Due on Tuesday 7 November 41 Let px dq be a metric space let p q P X Assume p q Show: DU P Bppq DV P Bpqq st U X V H 42 Let X be a topological space let W Ď X Assume P W DV P N X pqq st V Ď W Show that W is open in X

4 43 Let S T be topologies Let R : tu ˆ V U P S V P T u Show: xry fin X R 44 Show that N is discrete Recall: The topology on N is the stard topology T N pt q N Hint: You may use without proof the following two facts on X R S N Fact: Let S be a topological space Then: ( S is discrete ) iff P S tpu is open in S ) Focus Fact: Let X be a topological space let S Ď X let U Ď S Then: ( U is open in S ) iff ( Dopen V in X st U V X S ) 45 Let N : N Y t8u Show that N is NOT discrete Recall: The topology on N is the stard topology T N pt q N Hint: Use the same two facts from HW#44 but focus on X R S N U t8u Homework 8: Due on Tuesday 31 October 36 For all p P r1 8q let p be the p-norm on R 2 defined by px yq p r x p ` y p s 1{p Let 8 be the 8-norm on R 2 defined by px yq 8 maxt x y u Graph the unit spheres S 2 S 4 S 6 S 8 37 Let pv q be a normed vector space Let d be the stard metric on R defined by dpx yq x y Show for all x y P V that dp x y q ď d px yq 38 Let px dq be a metric space let q P X let s ą 0 Let B : Bpq sq Let p P B let r : s rdpp qqs Show that r ą 0 that Bpp rq Ď B 39 Let px dq be a metric space let p P X let U V P Bppq Show: ( U Ď V ) or ( V Ď U ) 40 Let R S be sets of sets P R DY P S st X Ď Y Show: Ť R Ď Ť S

5 31 Let V : R 3 let Homework 7: Due on Tuesday 24 October ε 1 : p1 0 0q ε 2 : p0 1 0q ε 3 : p0 0 1q Show ε 3 R xtε 1 ε 2 uy lin 32 Let V be a vector space let S Ď V Show that xsy lin is a subspace of V 33 Let V be a vector space let S Ď V let x P S Assume that x P xsztxuy lin Show that xsy lin xsztxuy lin 34 Show that tp1 0 0q p0 1 0qu is linearly independent in R 3 35 Find two bases B C of R 3 such that B X C H Homework 6: Due on Tuesday 17 October 26 Let S be a set of sets let X be a set Assume X P xsy Y P X DA P S st p P A Ď X 27 Let S be a set of sets let X be a set P X DA P S st p P A Ď X Show: X P xsy Y 28 Let V be a vector space let x y z P V Assume x ` z y ` z Show: x y 29 Let V be a vector space let a P R Show: a 0 V 0 V 30 Let V be a vector space let x P V Show: x 0 V x Homework 5: Due on Tuesday 10 October 21 Using the Principle of Mathematical Induction show: For all P N 1 2 ` ` 2 p ` 1qp2 ` 1q 6 22 ą 0 Dk P N st 1{k ă ε 23 z P R r z x ą 1 s ñ r Dy P Z st x ă y ă z s

6 24 z P P N z x ą 1 ñ k Dy P Z k st x ă y ă z 25 z P R r x ă z s ñ r Dy P Q st x ă y ă z s Homework 4: Due on Tuesday 3 October 16 Let n P N Define f : Z Ñ Z n by fpmq m n Show that f : Z ãñą Z n 17 Let S be a set Define Φ : t0 1u S Ñ 2 S by Φpfq f pt1uq Show that Φ : t0 1u S ãñą 2 S Hint: Define Ψ : 2 S Ñ t0 1u S by ΨpAq χ S A Show that Ψ Φ id t01u S that Φ Ψ id 2 S Use HW#14 18 Let S be a set Define f : S Ñ 2 S by fpzq tzu Show that f : S ãñ 2 S Note: The function f defined above is NOT onto 2 S so DON T try to prove that f : S ãñą 2 S 19 Let the function f : r1 8s Ñ r0 1s be defined by fpxq 1{x Show that f : r1 8s ãñą r0 1s # 1{x if x 0 Hint: Define g : r0 1s Ñ r1 8s by gpxq 8 if x 0 20 Show that Ť tr0 q P Nu r0 8q Homework 3: Due on Tuesday 26 September 11 Define h : R R by hpxq 1 ` 2 Show x 1 (1) dom rhs Rzt1u (2) h is onto Rzt2u 12 Define a P R N by a 2 Compute a 4 a 0 a 47 a 13 Define g : R R by gpxq? x ` 1 Find g pt0uq g pt0uq 14 Let A B be sets let f : A Ñ B let g : B Ñ A Assume that g f id A Show:

7 (1) f is 1-1 (2) g is onto A 15 Define f : R Ñ R by fpxq 3x ` 5 Show: for all y P R f 1 pyq py 5q{3 Homework 2: Due on Tuesday 19 September 6 ą 0 Dδ ą 0 y P R r p2 ă x ă 2 ` δq & p3 ă y ă 3 ` δq s ñ r 6 ă xy ă 6 ` ε s Hint: Remember that you can multiply inequalities provided all the terms are semipositive We calculate p2 ` δq p3 ` δq 6 ` 5δ ` δ 2 I suggest forcing 6 ` 5δ ` δ 2 ď 6 ` ε by forcing both 5δ ď ε δ 2 ď ε 2 2 by forcing both δ ď ε 10 " ε This suggests δ : min 10 c ε 2 * 7 ą 0 Dδ ą 0 y P R δ ď c ε 2 rp2 δ ă x ă 2 ` δq&p3 δ ă y ă 3 ` δqs ñ r5 ε ă x ` y ă 5 ` εs 8 Compute tx P Z x 2 ă 25u That is list all the elements in this set separate them by commas enclose the list in braces 9 Compute t1 2u Y t1 t1 2uu That is list all the elements in this set separate them by commas enclose the list in braces Note that one of the elements is the set t1 2u 10 For all n P N let A n : " x P R ˇˇ 1 n ă x ă 1 n Let B : Ş ta 1 A 2 A 3 u Using alternate notation Compute B B A 1 X A 2 X A 3 X *

8 Homework 1: Due on Tuesday 12 September 1 Let A B be propositions Show: p A ô B q ô p p A ñ B q & p B ñ A q q 2 Let A B C be propositions Show: r p A or B q ñ C s ô r p A ñ C q & p B ñ C q s 3 Let P Q be propositions Show: r P & p P ñ Q q s ñ Q 4 ą 0 Dδ ą 0 P R r 0 ă x ă 4δ s ñ r x ` 2x 2 ` 3x 5 ă ε s Hint: Force x ă ε 3 2x 2 ă ε 3 3x 5 ă ε 3 by forcing x ă ε x 2 ă ε x 5 ă ε by forcing x ă ε c c ε ε 0 ă x ă x ă by forcing 4δ ď ε 3 by forcing δ ď ε 12 by setting 4δ ď δ : min c ε 6 δ ď 1 c ε ą 0 Dδ ą 0 P R " c c ε 12 1 ε ε 4 9 c ε 4δ ď 5 9 δ ď 1 c 5 ε 4 9 r 2 δ ă x ă 2 ` δ s ñ r 15 ε ă 7x ` 1 ă 15 ` ε s *

9 Hint: Force 15 ε ă 7x`1 ă 15`ε by forcing 14 ε ă 7x ă 14`ε by forcing 2 ε 7 ă x ă 2 ` ε 7 by setting δ : ε 7