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 2 τ n \ (ii) If T1;:::Tn is a finite collection of members of τ, then Ti 2 τ i=1 [ (iii) If fTigi2I is any collection of members of τ, then Ti 2 τ i2I In this case, we call the pair (X; τ) a topological space and we call the sets T 2 τ open sets. Note, there are two topologies which we can always place on any set X: the trivial topol- ogy τ = f?;Xg 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 2 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 2 T , there Christian Parkinson GRE Prep: Topology & Real Analysis Notes 2 exists " > 0 such that (x − "; x + ") ⊂ T . Prototypical open sets in this topology are the open intervals (a; b) = fx 2 R : a < x < bg. Indeed, this interval is open because for x 2 (a; b), we can take " = minfjx − aj ; jx − bjg and we will find that (x − "; x + ") ⊂ (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] = fx 2 R : a ≤ x ≤ bg, 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) = fx 2 R : 0 ≤ x < 1g. 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; 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) = fx 2 R : a ≤ b < xg. 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 1 [ (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; 1) to be closed. However, both (−∞; a) and [b; 1) are easily seen to be open, so the set (−∞; a) [ [b; 1) 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 f?;Xg. 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: S2τ;S⊂T Sc2τ;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 2 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, B2β (2) if B1;B2 2 β, then for each x 2 B1 \ B2, there is B3 2 β such that x 2 B3 and B3 ⊂ B1 \ B2. Then the collection of sets τ = T : T = [i2I Bi for some collection of sets fBigi2I ⊂ 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 2 T , there is " > 0 such that (x − "; x + ") ⊂ 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 2 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 2 τ iff for all x 2 T , there is B 2 β such that x 2 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 2 β1, and for every x 2 B1, there is B2 2 β2 such that x 2 B2 and B2 ⊂ B1. (Be very careful not to mix up the inclusions in this statement.