Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios of elemets of T are i T A set for which a topology has bee specified is called a topological space A topological space is ofte deoted by the pair (X, T) wheever the specified topology is relevat The sets i the topology are called the ope sets A set E is called closed if c E is ope Notice that X ad the empty set are always both ope ad closed Examples (a) For ay set X the set of all subsets of X is a topology It is called the discrete topology The collectio {{}, X} is also a topology for ay set X It is called the trivial topology (b) Let T be the collectio of subsets U of X such that X U is either fiite or all of X The T is a topology ad it is called the fiite complimet topology To see that T is ideed a topology we eed oly to check the coditios of the defiitio Clearly, X ad the empty set are i T sice X X is empty (therefore, fiite) ad X {} is X If U T the X U is fiite Sice the itersectio of fiite sets is fiite, X U = X U ) is fiite Thus T If U T for i = 1, 2,, the X U i ( U is fiite Sice a fiite uio of fiite sets is fiite, X U i = ( X U i ) is fiite Thus U i T (c) Let X = { a, b, c} ad T = { X,{},{ a},{ b, c}} The T is a topology Note that {a} ad { b, c} are both ope ad closed (d) Let (X, T) be a topological space ad Y be a subset of X The collectio { Y U U T} is a topology o Y It is called the subspace topology (1) Check examples (c) ad (d) (2) Is it possible for a set to be ope i the subspace topology but ot ope i the space? Defiitios (a) A basis for a topology o X is a collectio B of subsets of X such that: (1) For each x X, there is at least oe B i B cotaiig x (2) If x B 1 B2 ( B 1, B 2 B) the there is a B 3 B such that x B3 B1 B2 (b) The sets i a basis are called basis elemets i
(c) A set U is ope i the topology geerated by B if for each x i U there is a basis elemet B cotaiig x such that B U (d) Suppose T 1 ad T 2 are topologies o the same set If T 1 T 2, we say that T 2 is fier tha T 1 ad that T 1 is coarser tha T 2 Note the directio of cotaimet; the fier topology cotais more ope sets tha the coarser topology If T 1 T 2 ad T 1 T 2, we say that T 2 is strictly fier tha T 1 ad that T 1 is strictly coarser tha T 2 (e) A space is Hausdorff if for each pair of poits x ad y i the space there are disjoit ope sets A ad B that cotai x ad y, respectively Examples (a) Suppose (X, T) ad (Y, T ) are topological spaces Let B = { U V U T ad V T } The B is a basis for a topology o X Y Ideed, every poit is i X Y which is itself a basis elemet ad the itersectio of two basis elemets is aother basis elemet ( ( U1 V1 ) ( U 2 V2 ) = ( U1 U 2 ) ( V1 V2 ) ) The product topology o X Y is the topology geerated by B (b) Suppose B 1 ad B 2 are bases for (X, T) ad (Y, T ), respectively The B = { B 1 B2 B1 B 1 ad B 2 B 2 } is a basis for the product topology o X Y (c) The stadard topology o R is the oe geerated by the basis {( a, b) a < b} The stadard topology o R 2 is the product topology of stadard topology o R with itself From (b) we see that {( a, b) ( c, d) a < b, c < d} is a basis for the stadard topology o R 2 (d) Ay simply ordered set ca be give the order topology It is the topology geerated by the basis {( a, b) a < b} {[ a0, b) a0 is the smallest elemet (if it exists)} {( a, b0 ] b0 is the largest elemet (if it exists)} O R the stadard topology is the order topology sice there is o largest or smallest elemet The positive itegers Z + is a ordered set with a smallest elemet The order topology o Z + is the discrete topology Theorems (a) Let B 1 ad B 2 be bases for topologies T 1 ad T 2, respectively The followig are equivalet: (1) T 2 is fier tha T 1 (2) For each elemet x ad each basis elemet B i B 1 cotaiig x, there is a basis elemet B' i B 2 such that x B' B (b) Suppose that C is a collectio of ope sets i a topological space X such that for each ope set U ad each x i U there is a C i C such that x C U The C is a basis for the topology o X (3) Show that the collectio B from example (b) is a basis
(4) Show that the collectio {( a, b) ( c, d) a < b, c < d ad a,b,c,d are ratioal}is a basis for the stadard topology o R 2 (5) Show that the collectio {[ a, b) a < b} is a basis o R (6) Show that the topology geerated by the basis from exercise (5) is strictly fier tha the stadard topology (7) Let B be a basis Show that the topology geerated by B equals the collectio of all uios of elemets from B (8) Show that every simply ordered set is Hausdorff i the order topology Fid a example of a o-hausdorff space (9) Fid examples that show that a ifiite itersectio of ope sets may be closed ad that a ifiite uio of closed sets may be ope A fuctio f : X Y betwee topological spaces is called cotiuous if for each ope subset O of Y, the set f 1 ( O ) is a ope subset of X We will see that this otio of cotiuity agrees with the usual otio of cotiuity from calculus Theorem Let f : X Y be a fuctio betwee topological spaces The the followig are equivalet: (a) f is cotiuous; (b) f ( E) f ( E) for every subset E of X; (c) f 1 ( C ) is closed i X wheever C is closed i Y Theorem (costructio of cotiuous fuctios) Let X, Y ad Z be topological spaces (a) The costat fuctio f ( x) = y 0 Y is cotiuous (b) If X is a subspace of Y the the iclusio map f ( x) = x Y is cotiuous (c) If f : X Y ad g : Y Z are cotiuous the g f : X Z is cotiuous (d) If f : X Y is cotiuous ad A is a subspace of X the the restricted fuctio f A : A Y is cotiuous (e) If f : X Y is cotiuous the a fuctio obtaied from f by restrictig or expadig the rage is cotiuous (f) The map f : X Y is cotiuous if X ca be writte as E where each is ope ad f E is cotiuous for each (g) The map f : X Y is cotiuous if for each x X ad each eighborhood V of f(x) there is a eighborhood U of x such that f ( U ) V This is called cotiuous at x (h) Let X = A B, where A ad B are closed; let f : A Y ad g : B Y be cotiuous fuctios such that f ( x) = g( x) for all x A B The the fuctio h : X Y defied by h ( x) = f ( x) for x A ad h ( x) = g( x) for x B is cotiuous E
(i) Let f : A X Y be give by the equatio f ( a) = ( f1( a), f 2 ( a)) The f is cotiuous iff f1 : A X ad f 2 : A Y are cotiuous Defiitios (a) A bijective fuctio f : X Y is called a homeomorphism if it ad its iverse are cotiuous (b) A mappig f : X Y is called a imbeddig if its restrictio f : X f ( X ) is a homeomorphism (c) A ope cover of a subset E of X is a collectio of ope sets E } from X such that E E (d) A set K is compact if every ope cover admits a fiite subcover More explicitly, if E } is ay ope cover of a compact set K the there is a fiite umber of sets { E 1,, E k from { E } such that E E1 Ek (e) Two disjoit oempty ope subsets A ad B of X are a separatio of X if A B = X If there is o separatio of X the it is called coected (f) The iterior of a set A is the largest ope set cotaied i A It is deoted by A o or It (A) (g) The closure of a set A is the smallest closed set that cotais A It is deoted by A (h) A poit x is a limit poit of a set A if every ope set cotaiig x itersects A at some poit other tha x The set of all limit poits of A is deoted by A (i) A eighborhood of x is ay set that cotais a ope set cotaiig x (j) A sequece { x } of poits coverges to a poit x if for every eighborhood U of x there is a positive iteger N such that x i U for all i > N If {x } does ot coverge, it diverges (k) The boudary of a set A is c A A It is deoted by Bd (A) (10) Show that ( X X X ) 1 2 1 X is homeomorphic to X 1 X 2 X (11) Show that the iterior of A is the uio of all ope subsets of A Show that the closure of A is the itersectio of all closed sets cotaiig A (12) Show that A = A A (13) Show that a set A is closed iff A = A ad that A is closed iff A A (14) Show that A o ad Bd (A) are disjoit ad that A = A o Bd (A) (15) If p is a limit poit of E (i a Hausdorff space) the every eighborhood of p cotais ifiitely may poits of E (16) A fiite poit set i a Hausdorff space has o limit poits (17) Show that if there is a sequece of poits from A covergig to x the x is a limit poit of A {
Theorems (coectedess) (a) X is coected iff the oly sets that are both ope ad closed are the empty set ad X itself (b) Aother formulatio of a separatio of X is a pair of oempty sets A ad B such that A B = X ad A B ad A B are empty (c) Suppose Y is a coected subspace of X If A ad B form a separatio of X the Y is etirely i either A or B (d) The uio of a collectio of coected sets with a poit i commo is coected (e) Let A be coected If A B A the B is also coected (f) The image of a coected set uder a cotiuous map is coected (g) If X i is coected for i = 1,, the the product i= 1 X i is coected (h) A subset E of the real lie is coected iff it has the followig property: If x, y E ad x < z < y, the z E (i) (Itermediate Value Theorem) Let f : X Y be a cotiuous map of a coected space X ito a ordered space Y (with the order topolog If a ad b are two poits of X ad r is a poit of Y lyig betwee f(a) ad f(b) the there is a poit c i X such that f(c) = r Theorems (compactess) (a) Every closed subset of a compact set is compact (b) If K is a compact set i a Hausdorff space ad x is ot i K the there exist disjoit ope sets A ad B such that K A ad x B (c) Every compact subset of a Hausdorff space is closed (d) The image of a compact space uder a cotiuous map is compact (e) Suppose X is compact ad Y is Hausdorff If f : X Y is a cotiuous bijectio the f is a homeomorphism (f) The product of fiitely may compact spaces is compact (g) Let X be a space with the order topology ad the least upper boud property Each closed iterval i X is compact (h) Let X be a oempty compact Hausdorff space If every poit of X is a limit poit of X the X is ucoutable Metric Spaces A set X is called a metric space if for ay two poits p ad q of X there is a associated umber d(p,q), called the distace from p to q, such that (a) d(p,q) 0; (b) d(p,q) = 0 iff p = q; (c) d(p,q) = d (q,p); (d) d(p,q) d(p,r) + d(r,q) for ay r i X Ay fuctio with these properties is called a distace fuctio or a metric
Defiitios (a) The set ( x, ε ) = { y d( x, < ε} is called the ope ball of radiusε cetered at B d x Similarly, the set C d ( x, ε ) = { y d( x, ε} is called the closed ball of radius ε cetered at x Whe o cofusio will arise the metric d will be omitted from the otatio (b) A set S is bouded if for every pair of poits x ad y i S there is a fiite umber M such that d( x, M A bouded set has a diameter that is defied to be the least upper boud of the set { d( x, x, y S } (c) Let : X Y be a sequece of fuctios from a set X to a metric space Y We Examples f say { f } coverges uiformly to f : X Y if give ε > 0 there is a iteger N such that d( f ( x), f ( x)) < ε for all > N ad all x i X (a) The usual distace fuctio d(p,q) = [ ( p i= 1 i q i ) 2 ] 1/ 2 (also called the Euclidea metric) turs R ito a metric space (b) Aother metric o the Euclidea spaces is give by d(p,q) = max{ p i q i for i = 1,,} This called the square metric (c) Let X be ay set ad d(p,q) be 1 if p q (ad, of course, 0 if p = q) The d is a metric (called the discrete metric) ad X is a metric space (d) Ay subset Y of a metric space X is a metric space usig the same distace fuctio If X has a metric d defied o it the the metric determies a topology (the metric topolog o X The collectio of all ope balls is a basis This basis geerates the metric topology o X All metric spaces are Hausdorff Differet metrics may geerate the same topology Example We show the Euclidea ad square metrics o R geerate the same topology Let d ad δ deote the Euclidea ad square metrics, respectively It is easy to see that δ ( x, d( x, δ ( x, holds for ay poits x ad y The first iequality shows that B d ( x, ε ) Bδ ( x, ε ) ad so the Euclidea metric topology is fier tha the square metric topology The secod iequality shows that Bδ ( x, ε / ) Bd ( x, ε ) ad so the square metric topology is fier tha the Euclidea metric topology Thus they are equivalet The topology geerated by both of these metrics is the same as the product topology o R (1) Show that the square metric topology is equivalet to the product topology o R
(2) If d is a metric, show that d ( x, = mi{ d( x,,1} is also a metric (called the stadard bouded metric associated to d) Show that all sets are bouded usig d (3) Show that i a Hausdorff space a coverget sequece must coverge to oly oe poit Example We show that the calculus ad topological otios of cotiuity agree o the real lie Suppose f : R R is cotiuous i the sese that give ε > 0 there is a δ > 0 such that f x) f ( x ) < ε wheever x x < δ Let O be ope ad x f 1 ( ) Sice f is ( 0 0 0 O cotiuous, x beig withi δ of x0 isures that f (x) is withi ε of f ( x 0 ) Choose ε 1 such that{ f ( x) : f ( x) f ( x0 ) < ε} O The ( x + δ, x δ ) f ( ) Thus f 1 ( O ) is ope Now suppose that 0 0 O f : R R is cotiuous i the sese that f 1 ( O ) is ope wheever O is ope The if O = f ( x ) ε, f ( x ) + ), f 1 ( O ) is ope Thus f 1 ( O ) cotais a ( 0 0 ε 1 ope ball of some radius δ > 0 cetered at x 0 Sice ( x0 δ, x0 + δ ) f ( O), x x 0 < δ implies that f ( x) f ( x0 ) < ε Theorems (a) Let X ad Y be metric spaces with metrics d X ad d Y, respectively The cotiuity of f : X Y is equivalet to the requiremet that give x adε > 0, there is a δ > 0 such that d ( x, < δ d ( f ( x), f ( ) < ε X Y (b) If X is a metric space the x A iff there is a sequece of poits i A that coverges to x (c) Suppose X is a metric space A fuctio f : X Y is cotiuous iff x x i X implies that f ( x ) f ( x) i Y (d) (Uiform Limit Theorem) If { f } is a sequece of cotiuous fuctios (from a topological space ito a metric space) that coverges uiformly to f the f is cotiuous (e) A subset of R is compact iff it is closed ad bouded i the Euclidea or square metric Defiitios (a) A space X is limit poit compact if every ifiite subset of X has a limit poit (b) A space X is sequetially compact if every sequece i X has a coverget subsequece (c) A fuctio f : X Y betwee metric spaces is uiformly cotiuous if giveε > 0, there is a δ > 0 such that for ay two poits a ad b of X, d ( a, b) < δ d ( f ( a), f ( b)) < ε X Y
Theorems (a) If E is a ifiite subset of a compact set K the E has a limit poit i K (ie, compactess limit poit compactess) (b) Limit poit compactess implies sequetial compactess i a metric space (c) Let A be a ope cover of a sequetially compact metric space X There is a δ > 0 such that for each subset of X havig diameter less tha δ there is a elemet of A cotaiig it (d) (Uiform Cotiuity Theorem) Let f : X Y be a cotiuous map from a compact metric space to a metric space The f is uiformly cotiuous (e) I a metric space X, compactess, limit poit compactess, ad sequetial compactess are equivalet Defiitios (a) A sequece {x } of poits i a metric space (X, d) is said to be a Cauchy sequece if giveε > 0, there is a iteger N such that d ( x, x m ) < ε wheever, m > N (b) A metric space is complete if every Cauchy sequece i the space coverges to a poit i the space (4) Is every coverget sequece a Cauchy sequece? (5) Is a closed set of a complete space complete? (If so, i what metric?) (6) If a space is complete uder d, is it complete uder d ( x, = mi{ d( x,,1}? Theorems (a) A metric space X is complete iff every Cauchy sequece i X has a coverget subsequece (b) Euclidea space R is complete i the Euclidea ad square metrics Let (Y, d) be a metric space ad d be the stadard bouded metric associated to d The collectio of all fuctios from some set X ito Y (deoted Y X ) ca be tured ito a metric space The formula ρ ( f, g) = lub{ d( f ( x), g( x)) x X} defies the uiform metric ρ o Y X correspodig to d If X is a topological space the we may cosider the collectio C (X,Y) of all cotiuous fuctios from X ito Y Theorems (a) If (Y,d) is complete the (Y X, ρ ) is complete (b) Uder the uiform metric, C (X,Y) is closed i Y X Thus, if (Y,d) is complete the (C (X,Y), ρ ) is complete (c) There exists a cotiuous surjectio f: [0,1] [0,1] 2
Baire Spaces A space X is said to be a Baire space if give ay coutable collectio {C } of closed sets i X each havig empty iterior the C also has empty iterior This defiitio may also be formulated i terms of ope sets: A space X is said to be a Baire space if give ay coutable collectio {U } of ope sets i X each beig dese i X the U is also dese i X (1) Show that these two defiitios are equivalet (2) Show that every o-empty ope set i a Baire space is ot the coutable uio of closed sets havig empty iteriors The followig statemet is kow as The Baire Category Theorem Every o-empty complete metric space is a Baire Space This is a very importat theorem i Aalysis It has a umber of cosequeces such as the Priciple of Uiform Boudedess, the Ope Mappig Theorem, the Closed Graph Theorem, the Iverse Mappig Theorem, ad the existece of a cotiuous owhere differetiable fuctio A subset F of C (X,R) is said to be uiformly bouded o a subset U of X if there is a positive iteger M such that f(x) < M for all x i U ad for all f i F Note: F beig bouded i ρ ( f, g) = lub { f ( x) g( x) : x X} is equivalet to F beig uiformly bouded o X Theorems (a) (Priciple of Uiform Boudedess) Let X be a complete metric space ad F be a subset of C (X,R) such that for each x i X the set F x = {f(x) f i F} is bouded The there is a oempty ope set U i X o which F is uiformly bouded (b) Let h:[0,1] R be a cotiuous fuctio Give ε > 0 there is a cotiuous owhere differetiable fuctio g:[0,1] R such that h(x) g(x) <ε for all x A set that ca be writte as a coutable itersectio of ope sets is called a Examples Gδ set (1) Every sigleto {x} i R is a Gδ sice {x}= B ( x,1/ ) =1 (2) Let f : R R be a arbitrary fuctio The set, A, cosistig of poits at which f is cotiuous is a G δ To see this let C be the collectio of all ope sets U such that the diameter of f(u) is less tha 1/ Set U to be the uio of all sets i C The U is ope ad A = U To see this equality we show double
cotaimet If x A the for every eighborhood V = ( f ( x) ε, f ( x) + ε ) of f (x) there is a eighborhood O = ( x δ, x + δ ) of x such that f ( O) V We show that x U Give, pick ( ( ) (4 ) 1, ( ) (4 ) 1 V = f x f x + ) The a eighborhood O of x such that f ( O ) V diameter( f ( O )) < (2) 1 < 1/ Thus x O U (O the other had) If x U the there exists a eighborhood O of x such that diameter ( f ( O )) < 1/ Give a eighborhood V = ( f ( x) ε, f ( x) + ε ) of f (x), pick > 1/ ε The diameter( f O )) < ε Thus Is there a fuctio because Q is ot a f ( O ) V x A ( f : R R that is cotiuous precisely o Q (the ratioals)? No, G δ To see this suppose Q is a G, ie, Q = W with each W ope If V q = R {q} (for q i Q ) the the collectio A = { W } { V q }is a coutable collectio of dese ope sets The, by the Baire Category Theorem, the itersectio A of all sets i A is also dese If x is i A the x is i W for each ad Vq for each q But this implies x is both i ad ot i Q Thus A is empty ad we have a cotradictio Is there a fuctio f : R R that is cotiuous precisely o the irratioals? Yes (See problem 16 from the metric space homework) This shows that the irratioals are a G δ (3) The Baire Category Theorem implies that R caot be writte as the coutable uio of closed sets havig empty iteriors Show that this fails if the sets are ot required to be closed (4) Show that the ratioals are ot Baire (5) Show that every ope subset of a Baire space is a Baire space δ