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Lecture 7 Outline Lecture 7 outline FUNCTIONS • Functions: A map ƒ: X → Y is a function. It assigns each element in X a unique element in Y. CONTINUITY • The notion of a function being continuous (with respect to a given topology) brings the open sets into the story as the open sets distinguish a certain subset of functions. • This subset of functions are called continuous. • Suppose X, Y are topological spaces: A function ƒ: X → Y is said to be continuous when the following occurs: If O ⊂ Y is any open set, then ƒ-1(O) is an open set in X. (Keep in mind that ƒ-1(O) is the set {x ∈ X: ƒ(x) ∈ O}. -1 • Graphical depiction of continuity and ƒ for a function from R to R. • Some spaces have no non-constant continuous functions to R: For example, R with the Zariski topology. • Let X denote a space with the property that any two open have non-empty intersection. Then there are no non-constant, continuous functions from X to a metric space. • This would be a sort of eternally frozen space because there is no measurable (i.e. temperature, pressure, humidity,…) to tell points apart. • Suppose that X is a non-Hausdorff space and p, q are points in X such that every open set that contains p also contains q. If ƒ is a continuous map from X to a metric space (R for example), then ƒ(p) = ƒ(q). There is no measurable way to tell p from q. RELATION TO CLASSICAL DEFINITION OF CONTINUITY • Classical definition of continuity: a) A function ƒ: R → R is continuous at x if, given a small ε > 0, there exists δ > 0 such that if |x - x´| < δ, then |ƒ(x´) - ƒ(x)| < ε. b) In the language of open sets: You specify an open set around ƒ(x) ((this is the set where |y - ƒ(x)| < ε). The function ƒ is continuous at x if I can guarantee that this ƒ(x´) will be in this set (thus |ƒ(x´) - ƒ(x)| < ε) if x´ is in a certain open set around x (this is the set where |x´ - x| < δ.) m n m n c) For maps from R to R : The same notion holds: A function/map ƒ: R → R is continuous at x if the following is true: You specify an open set around ƒ(x) (this is the set where |y - ƒ(x)| < ε). I can guarantee that ƒ(x´) will be in this set if x´ is in a certain open set around x (this is the set where |x´ - x| < δ.) • Mimic the last definition to define the notion of continuity for functions between a topological space X and a topological space Y: a) A function ƒ: X → Y is continuous at x if: You give me an open neighborhood of ƒ(x) (call it O) and I can give you an open neighborhood of x (call it U) such that if x´ ∈ U, then ƒ(x´) ∈ O. b) A function ƒ is continuous if it is continuous at every point. c) A function ƒ is continuous if ƒ-1(Open set) is an open set. (In this context, ƒ-1(O) is the set of points in X that map to O under ƒ.) d) Can substitute open sets with sets from a basis of open sets for topologies on Y and X. e) Why are these the same definition: To say that ƒ-1(O) is open is to say that if x maps by ƒ to a point in O, then there is an open set around ƒ that is also mapped by ƒ to O, and thus an open set around ƒ that entirely in ƒ-1(O). • Note that this is not the same as saying that ƒ(Open set) is open. For example if ƒ sends all of X to a point in R, which is a closed set, then ƒ is constant and it is certainly continuous. CONTINUOUS FUNCTIONS AND SEQUENCES • A sequence {xn}n=1,2,… converges in a topological space X to a point p if, given any open set O containing p, then there exists N such that {xn}n≥N is in O. Thus, all but finitely many are in O. • Another definition of continuity: A function ƒ is continuous if, given any convergent sequence {xn}n=1,2,… ⊂ X that converges to a point x, then {ƒ(xi)}i=1,2,… converges in Y to the point ƒ(x). • Questions: a) Are all functions that are continuous by this definition also continuous by the previous definition (ƒ-1(open set) is open)? b) Are all functions that are continuous by the open set definition continuous by this new definition? c) When are the two definitions equivalent? Assume that first that X and Y are Hausdorff spaces. d) (If a space isn’t a Hausdorff space, then a sequence can have many limits.) But then functions are not so useful in non-Hausdorff spaces (as we saw in the example above with the Zariski topology on (-∞, ∞). CONTINUOUS FUNCTIONS AND TOPOLOGIES • The set of continuous functions from X to Y depends on the topologies of X and Y. a) If T is coarser than T ´ (topologies for X) and ƒ is T-continuous, then it is T ´- continuous. b) If T is coarer than T ´ (topologies of Y) and ƒ is T ´ continuous, then it is T continuous. TOPOLOGIES ON X FROM FUNCTIONS ƒ: X → Y • Given a set X and ƒ: X → R, there is a topology that makes ƒ continuous: Take a -1 basis for the open sets {U : U = ƒ (O) with O being open in R}. This is the coarsest topology whereby ƒ is continuous. • Suppose ƒ: X → Y is a map with X being a set and Y being a topological space. a) The coarsest topology on X that makes ƒ continuous is as follows: A set U is open if and only if U = ƒ -1(O) with O ⊂ Y being open. I call this the induced topology. I can play this same game with any set of functions {ƒα: X → Xα}α∈J where the induced topology that makes them all continuous comes from the product topology on Πα∈J Xα and F: X → Πα∈J Xα given by taking the coordinate of F(X) in Xα to be ƒα(x). b) Metric topology: Let d: X × X → R be a distance function. The metric topology on X is the coarsest topology such that for every y ∈ X, the function that assigns to x the distance d(x, y) is continuous. c) Subspace topology: Let A denote a set and ƒ: A → X a map to a topological space. If ƒ is injective, then the subspace topology on A is the same as the induced topology for the map ƒ. • Embeddings: If X and Y are topological spaces and ƒ: X → Y is an injective, continuous map such that ƒ is a homeomorphism between X and ƒ(X) (the latter with the subspace topology), then ƒ is said to be an embedding. QUOTIENT TOPOLOGY: TOPOLOGIES ON Y FROM ƒ: X → Y. • Suppose that X has a topology but not Y. Let ƒ: X → Y denote a given map. The finest topology on Y that makes ƒ continuous is as follows: A set O ⊂ Y is open if and only if ƒ-1(O) is an open set in X. • Example of partitioning a set: Gluing polygons to get surfaces. This is an example of the quotient topology. • Constructing wormholes (ie. S1 × S2). • The 3-dimensional torus. .
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