Christian Parkinson GRE Prep: Topology & Real Analysis Notes 1
Week 6: Topology & Real Analysis Notes To this point, we have covered Calculus I, Calculus II, Calculus III, Differential Equations, Linear Algebra, Complex Analysis and Abstract Algebra. These topics probably comprise more than 90% of the GRE math subject exam. The remainder of the exam is comprised of a seemingly random selection of problems from a variety of different fields (topology, real analysis, probability, combinatorics, discrete math, graph theory, algorithms, etc.). We can’t hope to cover all of this, but we will state some relevant definitions and theorems in Topology and Real Analysis.
Topology
The field of topology is concerned with the shape of spaces and their behavior under continuous transformations. Properties regarding shape and continuity are phrased using the concept of open sets.
Definition 1 (Topology / Open Sets). Let X be a set and τ be a collection of subsets of X. We say that τ is a topology on X if the following three properties hold:
(i) ∅,X ∈ τ n \ (ii) If T1,...Tn is a finite collection of members of τ, then Ti ∈ τ i=1 [ (iii) If {Ti}i∈I is any collection of members of τ, then Ti ∈ τ i∈I In this case, we call the pair (X, τ) a topological space and we call the sets T ∈ τ open sets.
Note, there are two topologies which we can always place on any set X: the trivial topol- ogy τ = {∅,X} and the discrete topology τ = P(X). Having defined open sets, we are able to define closed sets.
Definition 2 (Closed Sets). Let (X, τ) be a topological space. A set S ⊂ X is called closed iff Sc ∈ τ. That is, S is defined to be closed if Sc is open.
The words open and closed can be a bit confusing here. Often times students mistak- enly assume that a set is either open or closed; that these terms are mutually exclusive and describe all sets. This is not the case. Indeed, sets can be open, closed, neither open nor closed, or both open and closed. In any topological space (X, τ), the sets ∅ and X are both open and closed. By De Morgan’s laws, since finite intersections and arbitrary unions of open sets are open, we see that finite unions and arbitrary intersections of closed sets remain closed.
Example 3. The set of real numbers R becomes a topological space with open sets defined as follows. Define ∅ to be open and define ∅ 6= T ⊂ R to be open iff for all x ∈ T , there Christian Parkinson GRE Prep: Topology & Real Analysis Notes 2 exists ε > 0 such that (x − ε, x + ε) ⊂ T . Prototypical open sets in this topology are the open intervals (a, b) = {x ∈ R : a < x < b}. Indeed, this interval is open because for x ∈ (a, b), we can take ε = min{|x − a| , |x − b|} and we will find that (x − ε, x + ε) ⊂ (a, b). We can combine open sets via unions or (finite) intersections to make more open sets; for example (0, 1) ∪ (3, 5) is also an open set. Likewise, prototypical closed sets are closed in- tervals [a, b] = {x ∈ R : a ≤ x ≤ b}, and any intersection or (finite) union of such sets will remain closed. As was observed above ∅ and R are both open and closed; in fact, in this space, these are the only sets which are both open and closed, though it is easy to construct sets which are neither open nor closed. Consider the set [0, 1) = {x ∈ R : 0 ≤ x < 1}. This set is not open because the point 0 is in the set, but it cannot be surrounded by an interval which remains in the set. The complement of this set is (−∞, 0) ∪ [1, ∞). This set is not open since 1 is in the set but cannot be surrounded by an interval which remains in the set. Since the complement is not open, the set [0, 1) is not closed. Note, this topology is called the standard topology on R.
Example 4. While the above example defines the standard topology on R, it is easy to come up with non-standard topologies as well. Indeed, let us now define T ⊂ R to be open if T can be written as a union of sets of the form [a, b) = {x ∈ R : a ≤ b < x}. These open sets comprise a topology on R. In this topology a prototypical open set is of the form [a, b). What other sets are open in this topology? Notice that
∞ [ (a, b) = [a + 1/n, b) n=1 which shows that sets of the form (a, b) remain open in this topology. Also notice that since [a, b) is open, we define [a, b)c = (−∞, a) ∪ [b, ∞) to be closed. However, both (−∞, a) and [b, ∞) are easily seen to be open, so the set (−∞, a) ∪ [b, ∞) is also open as a union of open sets. Since this set is open, it’s complement [a, b) is closed. Hence in this topology, all sets of the form [a, b) are both open and closed. The intervals [a, b] are closed and not open in this topology. Note, this topology is called the lower limit topology on R.
Notice in these example, the lower limit topology contains as open sets all of the sets which are open in the standard topology. In this way, the lower limit topology has “more” open sets and we can think of the lower limit topology “containing” the standard topology. We define these notions here.
Definition 5 (Finer & Coarser Topologies). Suppose that X is a set and τ, σ are two topologies on X. If τ ⊂ σ, we say that τ is coarser than σ and that σ is finer than τ.
On any space X, the finest topology is the discrete topology P(X) and the coarsest is the trivial topology {∅,X}. A finer topology is one that can more specifically distinguish between elements. Christian Parkinson GRE Prep: Topology & Real Analysis Notes 3
Definition 6 (Interior & Closure). Let (X, τ) be a topological space and let T ⊂ X. The interior of T is defined to be the largest open set contained in T . The closure of T is defined to be the smallest closed set containing T . We denote these by int(T ) and cl(T ) respectively. In symbols, we have [ \ int(T ) = S and cl(T ) = S. S∈τ,S⊂T Sc∈τ,T ⊂S
Other common notations are T˚ for the interior of T , and T for the closure of T .
Example 7. Considering R with the standard topology and a, b ∈ R, a < b, we have int([a, b)) = (a, b) and cl([a, b)) = [a, b].
In both of the examples above, there was some notion of a “prototypical” open set, from which other open sets can be built. We give this notion a precise meaning here.
Definition 8 (Basis (Base) for a Topology). Let X be a set and let β be a collection of subsets of X such that [ (1) X = B, B∈β
(2) if B1,B2 ∈ β, then for each x ∈ B1 ∩ B2, there is B3 ∈ β such that x ∈ B3 and B3 ⊂ B1 ∩ B2. Then the collection of sets τ = T : T = ∪i∈I Bi for some collection of sets {Bi}i∈I ⊂ β
forms a topology on X. We call this τ the topology generated by β, and we call β a basis for the topology τ.
This is half definition and half theorem: we are defining what it means to be a basis, and asserting that the topology generated by a basis is indeed a topology. If we can identify a basis for a topology, then the basis sets are the “prototypical” open sets, and all other open sets can be built as unions of the basis sets. Morally, basis sets are representatives for the open sets; if you can prove a given property for basis sets, the property will likely hold for all open sets. Often times it is easiest to define a topology by identifying a basis.
Example 9. Above we defined the standard topology on R by saying that a set T is open if for all x ∈ T , there is ε > 0 such that (x − ε, x + ε) ⊂ T . It is important to see this definition of the topology; however, this is a much more analytic than topological definition. The topological way to define the standard topology on R would be to define it as the topol- ogy generated by the sets (a, b) where a, b ∈ R, a < b. Indeed, these two definitions of the standard topology are equivalent as the following proposition shows. Christian Parkinson GRE Prep: Topology & Real Analysis Notes 4
Proposition 10. Suppose that (X, τ) is a topological space and that β is a basis for the topology τ. Then T ∈ τ iff for all x ∈ T , there is B ∈ β such that x ∈ B and B ⊂ T .
It is important to identify when two bases in a topological space generate the same topol- ogy and this next proposition deals with that question.
Proposition 11. Suppose that X is a set and β1, β2 are two bases for topologies τ1 and τ2. Then τ1 ⊂ τ2 iff for every B1 ∈ β1, and for every x ∈ B1, there is B2 ∈ β2 such that x ∈ B2 and B2 ⊂ B1. (Be very careful not to mix up the inclusions in this statement. What this is essentially saying is that β2 generates a larger (finer) topology iff β2 has more (smaller) sets.) Informally, the basis β2 generates a finer topology if we can squeeze basis sets from β2 inside basis sets from β1 (and not only that, but we can construct basis sets from β1 out of basis sets from β2).
Just as all groups have subgroups and all vector spaces have subspaces, there is a natural way to define subspaces of a topological space.
Definition 12 (Subspace Topology). Let (X, τ) be a topological space and let Y ⊂ X. Then the collection of sets σ = {Y ∩ T : T ∈ τ} forms a topology on Y . This topology σ is called the subspace topology on Y inherited from (X, τ).
Again, this is part definition and part theorem; we are asserting that such σ does indeed define a topology on Y .
Example 13. Consider [0, 3] ⊂ R with the standard topology on R. Note that the subspace topology on [0, 3] includes standard open sets like (1, 2) since this set is open in R and (1, 2) = [0, 3] ∩ (1, 2).
Now consider the set (1, 3]. This set is not open in R; however, it is open in the subspace topology on [0, 3], because (1, 4) is open in R and (1, 3] = [0, 3] ∩ (1, 4).
Likewise there is a natural way to combine topological spaces in a Cartesian product.
Definition 14 (Product Topology). Let (X, τ), (Y, σ). Recall the Cartesian product is given by coupling elements of X and Y : X × Y .= {(x, y): x ∈ X, y ∈ Y }. It is tempting to define a topology on X × Y comprised of sets of the form T × S for T ∈ τ, S ∈ σ. However, these do not form a topology on X × Y since a union of sets of this form will not be of this form anymore. So rather, we let β = {T × S : T ∈ τ, S ∈ σ} form the basis for a topology on X × Y . The topology generated by β is denoted τ × σ and the space (X × Y, τ × σ) is Christian Parkinson GRE Prep: Topology & Real Analysis Notes 5 the called the product space of X and Y .1
Example 15. Consider R with the standary topology, which we call τ1. The product space (R×R, τ1 ×τ1) can be visualized by drawing the plane with standard open sets as rectangles (a, b) × (c, d) = {(x, y) ∈ R × R : a < x < b and c < y < d}. Alternatively, we can consider 2 the set R of 2-dimensional vectors. On this space, we consider the topology τ2 generated 2 2 by circles: Br(v) = {z ∈ R : kv − zk < r} for v ∈ R and r > 0 (indeed, this is called the 2 x 2 standard topology on R ). We can identify each vector v = ( y ) ∈ R with the coordinates (x, y) ∈ R × R. Since this is a bijective map, the sets R × R and R2 are really the same. We’d like to know if the topologies τ1 × τ2 and τ2 are the same. To prove they are the same, consider the bases
β1 = {(a, b) × (c, d): a, b, c, d ∈ R, a < b, c < d} and x 2 β2 = Br(v): v = ( y ) ∈ R , r > 0 .
For any B1 = (a, b)×(c, d) ∈ β1, take any (x, y) ∈ B1 and let r = min{x−a, b−x, y−x, d−y}. x Then for v = ( y ), we will have x ∈ Br(v) ⊂ B1; this shows that for any (x, y) ∈ B1, we can find a set B2 ∈ β2 such that (x, y) ∈ B2 and B2 ⊂ B1. Hence by Proposition 11, τ1×τ1 ⊂ τ2. x z Conversely, let v = ( y ) and r > 0√ and consider the set B2 = Br(v) ∈ β2. For any u = ( w ) ∈ 0 0 0 0 0 Br(v), define r = (r − ku − vk2)/ 2. Then the square B1 = (z − r , z + r ) × (w − r , w + r ) z satisfies u = ( w ) ∈ B1 and B1 ⊂ B2. Thus by Proposition 11, we have τ2 ⊂ τ1 × τ1, and we can conclude that τ2 = τ1 × τ1. (Note, this inclusions of basis sets is pictured in Figure 1.) That is, the standard topology on R2 is the product of two copies of the standard topology on R. More generally, for n ∈ N, we can define the standard topology on Rn to be the topology generated by open balls and we will find that this is the same as the product of n copies of the standard topology on R.
Topology gives us the minimum structure required to discuss limits and continuity. In- deed, in calculus we were only able to discuss these things because R is naturally a topological space with the standard topology. We give the topological definitions of limits and continuity here and discuss some of their properties.
∞ Definition 16 (Limit of a Sequence). Let (X, τ) be a topological space and let {xn}n=1 ∞ be a sequence of values in X. We say that {xn}n=1 converges to a limit x ∈ X if for any open set T ∈ τ such that x ∈ T , there is N ∈ N, such that xn ∈ T for all n ≥ N. We write this as xn → x or lim xn = x. n→∞
1Note, this is only a good definition for the product topology when we are taking the product of a finite number of spaces. Indeed, if {(Xi, τi)}i∈I is an arbitrary collection of topological spaces, it is most natural Q to define the product topology on X = Xi to be the coarsest topology so that the projection maps i∈I Q πi : X → Xi are continuous. The topology defined generated by sets of the form i∈I Ui where Ui ∈ τi is then called the box topology. One can show that for a finite Cartesian product, the product topology and box topology agree with each other; this is not necessarily true for infinite products. (Another way to “correctly” Q define the product topology for an infinite product X = Xi is to let it be generated by sets of the form Q i∈I i∈I Ui where Ui ∈ τi and Ui = Xi for all but finitely many i ∈ I.) Christian Parkinson GRE Prep: Topology & Real Analysis Notes 6
Figure 1: A basis set from either topology τ1 × τ1 or τ2 can be fit around any point in a basis set from the other topology.
Note that this is a generalization of the definition we gave for a limit in calculus; in cal- culus we are always using the standard topology on R. One feature of the limit in calculus is that limits are unique: if xn → x and xn → y, then x = y. This is not true in a general topological space.
Example 17. Consider a space X with the trivial topology τ = {∅,X}. Take any sequence ∞ {xn}n=1 in X and apply the definition of the limit. For any x ∈ X, and any N ∈ N, we see that if T ∈ τ and x ∈ T , then T = X and xn ∈ T for all n ≥ N. Thus in this space, every sequence converges to every point.
There are non-trivial topologies where limits are still non-unique, but our intution tells us that limits should be unique and we can add a simple property to ensure that they are.
Definition 18 (Hausdorff Space). A topological space (X, τ) is called a Hausdorff space (or is said to have the Hausdorff property), if for all x, y ∈ X with x 6= y, there are open sets Tx,Ty ∈ τ such that x ∈ Tx, y ∈ Ty and Tx ∩ Ty = ∅.
Proposition 19. Limits in Hausdorff spaces are unique. That is, if (X, τ) is a Hausdorff ∞ ∞ space and {xn}n=1 is a sequence in X, then {xn}n=1 can have at most one limit x ∈ X.
Now we would like to discuss maps between spaces. As with linear transforms in linear algebra and homomorphisms in abstract algebra, we restrict our discussion to maps which preserve some of the underlying structure of the space. In topology, these are the continuous functions.
Definition 20 (Continuous Function). Let (X, τ) and (Y, σ) be two topological spaces. A function f : X → Y is said to be continuous iff f −1(V ) = {x ∈ X : f(x) ∈ V } ∈ τ whenever V ∈ σ. That is, f is continuous if the preimage of every open set in Y is open in X. Christian Parkinson GRE Prep: Topology & Real Analysis Notes 7
This gives us a way to define continuity without ever considering individual points. Con- trast this with the calculus definition of continuity where we first define what it means for a function to be continuous at a point, and then define continuous functions on a domain to be those which are continuous at each point. Of course, topology still has a notion of what it means to be continuous at a point.
Definition 21 (Pointwise Continuity). Let (X, τ) and (Y, σ) be two topological spaces and let f : X → Y . We say that f is continuous at the point x ∈ X, if for all V ∈ σ such that x ∈ V , we have f −1(V ) ∈ τ.
It is a good exercise to prove that when both X,Y are R with the standard topology, this definition of continuity is equivalent to the ε-δ definition of continuity presented in calculus.
Continuity is an important property because it preserves certain features of topological spaces. Indeed, if f : X → Y , we define the image of X under f by f(X) = {f(x): x ∈ X}. −1 We define σf to be the collection of subsets V ⊂ f(X) such that f (V ) is open in X. If f is continuous, this collection will form a topology, and this topology σf on f(X) will be the same as the subspace topology that f(X) inherits from (Y, σ). This shows that continuous functions respect the structure of the underlying spaces; in other words, continuous functions are the morphisms in the category of topological spaces. 2 Above we showed that the spaces (R , τ2) and (R × R, τ1 × τ1) are essentially the same space; it is important to be able to be able to identify equivalent spaces or distinguish be- tween distinct topological spaces and continuity helps us do that. We define a few more properties of topological spaces here.
Definition 22 (Connectedness). Let (X, τ) be a topological space and let C ⊂ X. We say that the set C is disconnected iff there exist non-empty A, B ∈ τ such that C ⊂ A ∪ B and A ∩ B = ∅. We say the space is connected iff it is not disconnected.
Example 23. Intuitively, a set is connected if it is in one whole piece; disconnected sets have separate pieces broken off from each other. Thus for exam- ple, in R with the standard topology, the set [5, 9) is connected while the set {1} ∪ [5, 9) ∪ (10, ∞) is dis- connected. However note, this is only a heuristic! It works very well in R, but even in R2 there are fa- mous examples that challenge this intuition. Indeed, consider the set C ⊂ R2 (pictured right) given by
1 C = {(0, y): y ∈ [−1, 1]}∪ x, sin x : x ∈ (0, ∞) . This is called the Topolgist’s Sine Curve, and while it is defined as the disjoint union of two sets, it is actu- ally connected. Indeed, any open set containing the Figure 2: Topologist’s Sine Curve vertical strip {(0, y)}y∈[−1,1] will necessarily contain Christian Parkinson GRE Prep: Topology & Real Analysis Notes 8 some of the curve. Thus this set cannot be covered by two disjoint, open sets. (Actually, a stronger notion of connectedness is path-connectedness. Roughly speaking, a set is path-connected if one can draw a path between any two points in the set without leaving the set. At first glance, path-connectedness may seem equivalent to connectedness, but the topologist’s sine curve is connected without being path-connected.)
One important result involving connectedness helps us classify sets which are both open and closed.
Proposition 24. Suppose that (X, τ) is a topological space. Then X is connected iff the only sets which are both open and closed in X are ∅ and X itself.
Definition 25 (Compactness). Let (X, τ) be a topological space and let C ⊂ X. We say that C is compact iff from any collection of open sets {Ui}i∈I such that C ⊂ ∪i∈I Ui, we n can extract a finite collection of sets Ui1 ,...,Uin such that C ⊂ ∪k=1Uik .
Such a collection {Ui}i∈I in the definition of compactness is called an open cover of C. Thus, in words, a set C is compact if every open cover of C admits a finite subcover.
Example 26. Compactness is somehow a generalization of closedness and boundedness. Indeed, we give examples here of a bounded set which is not closed and a closed set which is not bounded and prove that neither are compact. Consider R with the standard topol- ogy and consider the bounded set (0, 1). This set is not closed, and we show that it is not ∞ compact. Consider Uk = (0, 1 − 1/k) for k ∈ N. We see that (0, 1) ⊂ ∪k=1Uk so {Uk} forms an open cover of (0, 1). However, if we take any finite subcollection Uk1 ,...,UkN of {Uk} N and let K = max{k1, . . . , kN }. Then (0, 1) 6⊂ ∪n=1Ukn = (0, 1 − 1/K). Thus there cannot be a finite subcover for this particular cover, and so this cover violates the definition of compactness and we conclude that (0, 1) is not compact. Similarly, consider the set [0, ∞). This set is closed, but is not bounded, and we show that it is not compact. Consider the ∞ collection Uk = (−1, k) for k ∈ N. We see [0, ∞) ⊂ ∪k=1Uk, but again, any finite collection N Uk1 ,...,UkN will satisfy [0, ∞) 6⊂ ∪n=1Ukn = (−1,K) where K = max{k1, . . . , kN }, and so this open cover admits no finite subcover, and hence we conclude that (0, ∞) is not compact. By contrast, a closed and bounded subset of R like [0, 1] is compact, though this is not at all trivial to prove. In general topological spaces, it is easier to show that a set isn’t compact, since this only requires exhibiting one example of an infinite cover that does not admit a finite subcover. In R (and more generally, in metric topologies), there are nice theorems which give concrete lists of properties which are equivalent to compactness. We cover some of these theorems in the Real Analysis portion of these notes.
Proposition 27. Suppose that (X, τ) and (Y, σ) are topological spaces and that f : X → Y is continuous. Let U ⊂ X and consider the image of U under f defined by f(U) .= {f(x): x ∈ U} ⊂ Y . If U is connected in X, then f(U) is connected in Y . Likewise, if U is compact, then f(U) is compact in Y .
This proposition tells us that these properties of connectedness and compactness are in- Christian Parkinson GRE Prep: Topology & Real Analysis Notes 9 variant under continuous maps. Statements like this help us characterize topological spaces. Note however that the converse is not true: a continuous map could still map a disconnected set to a connected set (for example, the continuous function f(x) = x2 on R maps the dis- connected set (−1, 0) ∪ (0, 1) to the connected set (0, 1)), or a non-compact set to a compact set (for example, the continuous map f(x) = sin(x) maps the non-compact set (0, ∞) to the compact set [−1, 1]). If we want a continuous map to not change the structure of a space at all, we need to require something more.
Definition 28 (Homeomorphism). Suppose that (X, τ) and (Y, σ) are topological spaces. A function f : X → Y is called a homeomorphism iff the following four properties hold:
1. f is one-to-one,
2. f is onto,
3. f is continuous,
4. f −1 is continuous.
If such a function f exists, the topological spaces (X, τ) and (Y, σ) are called homeomorphic and we write X ∼= Y .
In words, a homeomorphism between two topological spaces is a bicontinuous bijection; it maps each space bijectively and continuously to the other. Homeomorphic spaces share essentially all important properties in common, so when two spaces are homeomorphic we think of them as ”morally the same space.” For this reason it can be important to identify whether two spaces are homemorphic. We make a few final statements about properties that homeomorphic spaces share, and give one example in conclusion.
Proposition 29. Let (X, τ) and (Y, σ) be topological space, let U ⊂ X and let f : X → Y be a homeomorphism. Then
U is open in X iff f(U) is open in Y ,
U is closed in X iff f(U) is closed Y ,
U is connected in X iff f(U) is connected in Y ,
U is compact in X iff f(U) is compact in Y ,
X is Hausdorff iff Y is Hausdorff.
Example 30. Consider R with the usual topology. Any open interval (a, b) is homeomorphic to the interval (0, 1) under the map f :(a, b) → (0, 1) defined by x − a f(x) = , x ∈ (a, b). b − a Christian Parkinson GRE Prep: Topology & Real Analysis Notes 10
Likewise, (0, 1) is homeomorphic to R under the map g : (0, 1) → R defined by