notes3_topological spaces

Chapter 3 Topological Spaces 3.1 Denition and Some Examples We want to generalize the concepts that we developed in studying the metric spaces. We want to remove our reliance on a distance function. We were able to dene most of what we wanted to do in metric spaces by dening our concepts in terms of the open sets. This was especially true of our study of continuous functions. We will use the results that we proved about open sets as our basis for the gener- alization. We will dene open sets as sets that satisfy certain conditions. Denition 3.1 LetXbe a set and Ta family of subsets of Xsatisfying the following properties. a) The set Xand;belong to T, b) The union of any family of members of Tis a member of T. c) The intersection of any nite family of members of Tis a member of T. ThenTis called a topology forXand the members of Tare called open sets . The ordered pair (X;T)is called a topological space , or simply a space. If we use the terminology open sets instead of member of T, then the denition of a topological space may be restated as follows: A family of subsets of Xis a topology forXmeans that: a) BothXand;are open sets. b) The union of any family of open sets is an open set. c) The intersection of any nite family of open sets is open. 23 24 CHAPTER 3. TOPOLOGICAL SPACES Example 3.1.1 The usual topology for the real line Ris the topology generated by its usual metric. We shall refer to the real line with the usual topology as simply the real line orR. Example 3.1.2 The usual topology onRnis the topology generated by the usual metric onRn. It is also the topology generated by the taxicab metric and the max metric. Thus, the usual topology does not distinquish the metric determining it from the other two. We shall refer to Rnwith the usual topology as Euclidean n-space , or simplyRn. Example 3.1.3 For any set Xwe take T= 2Xto be the set of all subsets of X. This clearly satises all of the properties of a topology, since we have included every possible subset in the topology. This is called the discrete topology . Note that it is the topology generated by the discrete metric. Also, note that this is the largest possible collection of open subsets of X. Example 3.1.4 At the opposite extreme, we may take T=f;;Xg. This is called thetrivial topology , or indiscrete topology , onX. This is the smallest collection of open sets on X. Example 3.1.5 LetXbe a set. We shall take Tto consist of;,X, and all sets U so thatXnUis a nite set. Then Tis a topology on Xcalled the conite topology , ornite complement topology . This is really of interest only when Xis an innite set. WhenXis a nite set, this is the same as the discrete topology. Denition 3.2 A subsetFof a topological space Xisclosed ifXnFis an open set. Theorem 3.1 The closed sets of a topological space Xhave the following properties: a)Xand;are closed. b) The intersection of any family of closed sets is closed. c) The union of any nite family of closed sets is closed. Denition 3.3 Let(X;T)be a topological space and let AX. A pointxinXis alimit point ofAif every open set containing xcontains a point of Adistinct from x. The set of limit points of Ais called the derived set ofA, denotedA0. c 1999, David Royster Introduction to Topology For Classroom Use Only 3.2. INTERIOR, CLOSURE AND BOUNDARY 25 Example 3.1.6 LetX=fa;b;c;dg. LetT0be the indiscrete topology; T1, the discrete topology; T2=f;;X;fag;fbg;fa;bgg; and T3=f;;X;fag;fbg;fcg;fa;bg;fa;cg;fb;cg;fa;b;cgg. The reader should verify that T2andT3are topologies on X. LetA=fa;bg,B=fcg, andC=fdg. We want to nd the limit points of these sets in the dierent topologies. T0T1T2T3 AX;fc;dgfdg Bfa;b;dg;fdgfdg Cfa;b;cg;fcg; Theorem 3.2 A subsetAof a topological space Xis closed if and only if Acontains all of its limit points. This is no surprise, and is proven exactly the way in which we proved it earlier in a metric space. We were careful there not to use the distance function, but to use the open sets. Denition 3.4 LetXbe a topological space and let fxngbe a sequence of points inX. We say thatfxngconverges to the point x2X, orxis the limit of the sequence, if for each open set Ucontaining xthere is a positive integer Nso that xn2Ufor allnN. Sequences are not as fundamental in general topological spaces as they are in metric spaces. The following example may show why. Example 3.1.7 ConsiderRwith the conite topology. Let fxngbe any sequence of real numbers. Let a2Rbeany real number. Then fxngconverges to a, because if Uis any open set containing a, thenRnUis a nite set. Since fxngis an innite set, we must have that innitely many members of fxnglie inU. Thus, there is a positive integer Nsuch that if nN xn2U. Thus,fxngconverges to a. However, awas any arbitrary real number. This means that fxngconverges to every real number . What is more (and maybe worse) is that fxngwas an arbitrary sequence. This means that every sequence converges to every real number. There are no non- convergent sequences and sequences do not have unique limits. 3.2 Interior, Closure and Boundary Just as before we will dene the interior, closure, and the boundary. Denition 3.5 LetAbe a subset of the topological space X. A pointx2Ais an interior point ofAif there is an open set Uso thatx2UA.Ais called a neighborhood ofx. The interior ofA, denoted by A, is the set of all interior points ofA. c 1999, David Royster Introduction to Topology For Classroom Use Only 26 CHAPTER 3. TOPOLOGICAL SPACES Theclosure ,A, of A is the union of Aand it set of limit points: A=A[A0: A pointx2Xis aboundary point ofAifx2A\XnA. The set of boundary points ofAis called the boundary ofAand is denoted by @A. Theorem 3.3 For any subsets A;B of a topological space X a) The interior of Ais the union of all open sets contained in Aand is the largest open set contained in A. b)Ais open if and only if A=A. c) IfAB, thenAB. d)(A\B)=A\B. Proof: We will oer only a proof for (d). The others follow from closely from what we did in the case of a metric space. SinceA\Bis a subset of both AandB, then by (c) ( A\B)A\B. Now, A\Bis an open set and is a subset of A\B. Thus by (a), A\B(A\B). This completes the proof. Theorem 3.4 For any subsets A;B of a topological space X a) The closure of Ais the intersection of all closed sets containing in A and is the smallest closed set containing in A. b)Ais closed if and only if A=A. c) IfAB, thenAB. d)(A[B) =A[B. We leave this to the reader to prove. Theorem 3.5 LetAbe a subset of a topological space X. a)@A=A\XnA=@(XnA). b)@A,A, and (XnA)are pairwise disjoint sets whose union is X. c)@Ais a closed set. d)A=A[@A. e)Ais open if and only if @A(XnA). c 1999, David Royster Introduction to Topology For Classroom Use Only 3.3. BASIS FOR A TOPOLOGY 27 f)Ais closed if and only if @AA. g)Ais open and closed if and only if @A=;. Proof: Parts (a){(d) follow immediately from the denitions. (e) IfAis open then A=A. Now by (b) Aand@Aare disjoint. Thus, Aand @Aare disjoint. This implies that @AXnA. Now, if@AXnAthen no point ofAis a boundary point of A. Thus, every point of Ais an interior point of Aand A=A. Thus,Ais open. (f) This follows from our duality of open and closed sets. (g) IfAis both open and closed, then @AA\(XnA) =;. If@A=;then clearly@AA| meaning Ais closed | and @A(XnA) | meaning Ais open. Denition 3.6 A setAin a topological space Xisdense ifA=X. IfXhas a countable dense set, then Xis aseparable space. Example 3.2.1 1. The reals with the usual topology is separable, since the ra- tionals are dense. 2. Euclidean n-space is separable, since the set of points having only rational coordinates is dense and countable. 3. The reals with the conite topology is separable, since every countable innite set is dense. Denition 3.7 A subsetBof a spaceXisnowhere dense if(B)=;. Note that a nite subset of a metric space is nowhere dense. In other topological spaces, we will see more interesting examples. 3.3 Basis for a Topology It appears that a topology can be relatively large. In fact, for an innite set the discrete topology consists of all subsets of the space, so it would be prohibitive to have to check all subsets. We have seen though that we can get by with just checking some of the sets. For the discrete topology we have usually only checked the singleton sets. For a metric space we were able to do everything we wanted by working with the open balls. In fact we dened all open sets in terms of the open balls. Can we do this in general? Can we nd a certain collection of subsets that will generate all of the elements of the topology, just like the open balls generate the metric topology? The answer is yes, because we can take Tas this generating set. This begs the answer, because we are looking for a smaller collection than the whole topology. c 1999, David Royster Introduction to Topology For Classroom Use Only 28 CHAPTER 3. TOPOLOGICAL SPACES Denition 3.8 Let(X;T)be a topological space. A basisBforTis a subcollection ofTwith the property that each member of Tis a union of members of B. The members of Bare called basic open sets andTis the topology generated by B. Example 3.3.1 Most of what we have seen is based on metric spaces. 1. The collection of all open intervals is a basis for the usual topology on the reals. 2. The collection of all open balls is a basis for the metric topology on the metric space (X;d). 3. For any set Xthe collection of all singleton sets fxgis a basis for the discrete topology. Denition 3.9 Let(X;T)be a topological space. A local basis ata2Xis a subcollection BaofTsuch that a)abelongs to each member of Ba, and b) each open set containing acontains a member of Ba. Denition 3.10 A spaceXisrst countable if there is a countable local basis at each point of X. The space Xissecond countable if the topology for Xhas a countable basis. Note that every second countable space is rst countable because if there is a countable basis B, then the number of these sets containing any given point a2X is at most countable. Theorem 3.6 Every second countable space is separable. Proof: LetXbe a second countable space with a countable basis B. LetAbe a set formed by choosing one element from each non-empty element of B. Each point ofXis a limit point of some point in Aby the denition of a basis. Thus, Ais dense inX. Theorem 3.7 a) Every metric space is rst countable. b) Every separable metric space is second countable. The proof is left to the reader. We have been starting with a topology and asking if there is a basis for it. We could be starting with a collection of open sets and asking if it forms a basis for a topology. Not every collection of open sets will work. When is a collection of open sets a basis for a topology on X? c 1999, David Royster Introduction to Topology For Classroom Use Only 3.4. CONTINUOUS FUNCTIONS 29 Theorem 3.8 A family Bof subsets of a set Xis a basis for a topology on Xif and only if both of the following hold: a) The union of members of BisX. b) For each B1;B22Bandx2B1\B2, there is a member BxofBsuch thatx2BxB1\B2. Example 3.3.2 [Sorgenfrey Line] Let Bbe the collection of all half-open intervals ofRof the form [ a;b),a < b . Clearly, the union of all of these intervals is R. If we take two of these sets and intersect them, we can nd another of these sets in the intersection. Thus, these sets form a basis for a topology on the real line, called the half-open interval topology T00forR.Rwith this topology is called the Sorgenfrey line. The Sorgenfrey line has the property that it is rst countable and separable, but not second countable. 3.4 Continuous Functions We were able to move away from the epsilon-delta denition of continuity of a function in between two metric spaces by using open balls. We will use this as our starting point for general topological spaces. Denition 3.11 A function f: (X;T)!(Y;T0)iscontinuous at a pointa2X if for each open set VinYcontainingf(a)there is an open set UXcontaininga so thatf(U)V, or equivalently Uf1(V). Now, we are more interested in the situation where the function is continous at every point of X. This means that for every open set VinYand every point a2X withf(a)2V, there is an open set UaXwitha2Uaandf(Ua)V. Equivalently, we havea2Uaf1(V). This means that for each point in f1(V) we can nd an open set containing that point and contained in f1(V). Thus,f1(V) must be open inXfor each open set VY. This leads us to a more general denition. Denition 3.12 A function f: (X;T)!(Y;T0)iscontinuous if for each open setVinY f1(V)is an open set in X. Theorem 3.9 Letf:X!Ybe a function on the topological spaces XandYand leta2X. The following are equivalent. a)fis continuous at a. b) For each open set V2Ycontaining f(a), there is an open set UinX such thata2Uf1(V). c) For each neighborhood Voff(a),f1(V)is a neighborhood of a. c 1999, David Royster Introduction to Topology For Classroom Use Only 30 CHAPTER 3. TOPOLOGICAL SPACES The proof is left to the reader. Theorem 3.10 Letf:X!Ybe a function of topological spaces. The following are equivalent. (i)fis continuous. (ii) For each closed subset CY,f1(C)is closed in X. (iii) For each subset AX,f(A)f(A). (iv) There is a basis Bfor the topology of Yso thatf1(B)is open inXfor each basic open set B2B. Proof: To show that ( i) implies ( ii) will require the duality between open and closed sets. IfCYis closed, the YnCis open inY. Sincefis continuous, f1(YnC) is open inX. HenceXn(f1(YnC)) is open. If x2Xn(f1(YnC)) thenf(x)62YnC, orf(x)2C. Thus,Xn(f1(YnC))f1(C). The opposite inclusion is clear. Thus, f1(C) =Xn(f1(YnC)) is closed. A similar analysis shows that ( ii) implies ( i). To show that ( ii) implies ( iii), letAX. Thenf(A) is a closed subset of Y. Hence,f1(f(A)) is a closed subset of X. Now,Af1(f(A)) soAf1(f(A)). Thus,f(A)f(A). To show that ( iii) implies ( ii), letCbe a closed subset of Y. Then, f(f1(C))ff1(C)CC sof1(C)f1(C) makingf1(C) a closed set. For the last equivalence, ( i) clearly implies ( iv). We need to prove the opposite implication. Let Obe an open set in Y. Then by the denition of a basis, O=[2IB for some subcollection fBg2Iof the basis B. Then f1(O) =f1 [ 2IB! =[ 2If1(B): Since each f1(B) is open in Xand the union of any family of open sets is open, thenf1(O) is open in Xandfis continuous. Theorem 3.11 Iff:X!Yandg:Y!Zare continuous functions, then g f:X!Zis continuous. Denition 3.13 A function f:X!Yis ahomeomorphism if a)fis one-to-one, ( injective ) b)fis onto, ( surjective ) c 1999, David Royster Introduction to Topology For Classroom Use Only 3.5. SUBSPACES 31 c)fis continuous, d)f1is continuous. Topological spaces are topologically equivalent orhomeomorphic if there home- omorphism from ffromXontoY. The continuity of the function and its inverse are extremely important! A prop- ertyPof topological spaces is a topological property ortopological invariant provided that if space Xhas property P, then so does every space which is homeo- morphic to X. Theorem 3.12 Separability is a topological property. Theorem 3.13 First countability and second countability are topological properties. Denition 3.14 A topological space is metrizable provided that the topology on X is generated by a metric. Theorem 3.14 Metrizability is a topological property. SinceRand (0;1) are homeomorphic, the property of being a bounded metric space is not a topological property. Likewise, distance is not a topological invariant. 3.5 Subspaces Let (X;T) be a topological space and let Abe a subset of X. The relative topology orsubspace topology T0onAdetermined by Tconsists of all sets of the form O\Afor whichOis an open set of T. T0=fO\AjO2Tg: The members of T0are called relatively open sets inA, and (A;T0) is called a subspace of (X;T). Note that this is actually a topology for A. ;=;\A A =X\A; so both;andAare open in A. IffUgare open in A, thenU=O\Aand [ U=[ (O\A) = [ alphaO! \A is relatively open since the union of any family of open sets is open in X. For any nite family of open sets Ui=Oi\A, we have n\ i=1Ui=n\ i=1(Oi\A) = n\ i=1Oi! \A is relatively open since the intersection of any nite family of open sets is open in X. c 1999, David Royster Introduction to Topology For Classroom Use Only 32 CHAPTER 3. TOPOLOGICAL SPACES Lemma 3.1 Let(a;T0)be a subspace of the topological space (X;T). A subset F ofAis closed in the subspace topology on Aif and only if F=C\Afor some closed subsetCofX. Example 3.5.1 1. The closed interval [ a;b] witha < b is a subspace of Rwith the usual topology. The open sets containing aare sets of the form [ a;c) with a<c<b . 2. The subset of Rn+1consisting of all ( n+ 1)-tuples ( x1;x2;::: ;xn;xn+1) with xn+1= 0 is homeomorphic to Rn 3. Leta<b<c<d . LetA= [a;b][(c;d) be considered as a subspace of the real line. Then the subset [ a;b] ofAis both relatively open and relatively closed. It is clearly closed because [ a;b] = [a;b]\Aand [a;b] is closed in the real line. It is open because for 0 <  < cb, [a;b] = (a;b+)\Y. Thus, we see that since ( c;d) is the complement of this set that is both relatively open and relatively closed, then we see that ( c;d) is both relatively open and relatively closed. Denition 3.15 A property Pto topological spaces is hereditary provided that if Xhas property P, then every subspace of Xhas this property. Example 3.5.2 1. First countability and second countability are hereditary prop- erties. IfXhas a countable basis, then intersecting these basis elements with A will give a countable basis for the subspace topology. First countable is similar. 2. Separability is not hereditary. Let AR2consist of the x-axis and the point a= (0;1). Dene a topology TonAby taking the empty set and all subsets ofAthat contain the singleton set fag. Then (X;T) is separable because the singleton setfagis dense. Every point except ais a limit point offag. However, the subspace topology on R(thex-axis) is the discrete topology, so ( R;T0) is not separable. 3.6 Hausdor Spaces A topological space Xis aHausdor space if for each pair of distinct points a;b2Xthere exist disjoint open sets UandVsuch thata2Uandb2V. Example 3.6.1 1. Every metric space is Hausdor. We proved this as a home- work problem, but it is simple. Let r=d(a;b) and then take two open balls of radiusr=2 centered at aandbrespectively. 2. The real line with the co-nite topology is not Hausdor. Likewise, the real line with the co-countable topology is not Hausdor. c 1999, David Royster Introduction to Topology For Classroom Use Only 3.6. HAUSDORFF SPACES 33 3. Take any set with more than one point and give it the indiscrete (trivial) topol- ogy. It is not Hausdor. 4. The space in Example 2 is not Hausdor, because you can never separate a from any other point. 5. [The Zariski Topology ] Letnbe a positive integer and consider the family P of all polynomials in nreal variables x1;x2;:::;xn. Forp2PletZ(p) denote its solution set in Rn: Z(p) =f(x1;x2;:::;xn)2Rnjp(x1;x2;:::;xn) = 0g: LetBbe the collection of sets that are complements of some Z(p) for some p2P. This forms a basis for a topology on Rncalled the Zariski topology . For the real line, n= 1, this is just the co-nite topology. This is because each nite set of real numbers is the solution set for some polynomial in one real variable. If A=fa1;a2;:::;ang, then p(x) = (xa1)(xa2):::(xan) is a polynomial with Aas its solution set. Likewise, the set of solutions of a polynomial in one real variable of dimension nis at mostn. Forn>1 this is not the co-nite topology. For example, the line y=ainR2 is the solution set to the polynomial in two variables p(x;y) =ya: Note that this is not a nite set. However, each nite set can serve as the solution set of a polynomial. Now,Rnwith the Zariski topology is not Hausdor. Assume that P= (a1;a2;:::;an) andQ= (b1;b2;:::;bn) are two distinct points in Rn. Theorem 3.15 1. The property of being a Hausdor space is topological and hered- itary. 2. In a Hausdor space a sequence fxng1 n=1cannot converge to more than one point. Proof: We will prove 1 only and leave the other to the reader. Suppose that Xis Hausdor and f:X!Yis a homeomorphism. For a6=b2Y, we have that f1(a) andf1(b) are distinct points in X. Thus, there are disjoint open setsU;VXso thatf1(a)2Uandf1(b)2V. Hence,f(U) andf(V) are disjoint open sets of Ycontainingaandbrespectively. To show that Hausdor is hereditary, assume that Xis Hausdor and that AX. Leta6=b2A. Then,a6=b2Xand there are disjoint open sets U;VXso that a2Uandb2V. Then,U\AandV\Aare disjoint relatively open subsets of A containingaandbrespectively. c 1999, David Royster Introduction to Topology For Classroom Use Only