Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Surface (topology) wikipedia , lookup
Orientability wikipedia , lookup
Sheaf (mathematics) wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Covering space wikipedia , lookup
Fundamental group wikipedia , lookup
Continuous function wikipedia , lookup
TOPOLOGY ASSIGNMENT 6 CONTINUOUS FUNCTIONS, HOMEOMORPHISMS. PRODUCT SPACES. FOR SUBMISSION: Q2-5, DUE NOVEMBER 18 IGOR WIGMAN (1) Using your results in Assignment 5 Q4 decide for each of the functions of Assignment 5 Q4 whether it is a homeomorphism. S (2) Let {Ai }i∈I be a collection of subsets of a topological space X, such that X = Ai . Let i∈I f : X → Y be a map of topological spaces such that the restriction of f to each Ai (equipped with the subspace topology induced from X) is continuous. (a) Show that if each Ai is closed, and I is finite, then f is continuous. (b) Find an example where I is countable and each Ai is closed, but f is not continuous. (c) Find an example where |I| = 2 (that is, X = A1 ∪A2 ), A1 is closed, but f is not continuous. (3) (a) Prove or disprove: Any same cardinaly sets X, Y , equipped with the finite complement topology Tf are homeomorphic. What about the countable complement topology Tc ? (b) Let f : Rl → R be a function. Prove that f is continuous, if and only if it is right continuous at all x ∈ R (i.e. lim f (x) = f (x0 )). Find a condition for f to be continuous as a funcx→x0 + tion R → Rl , and Rl → Rl . Q (4) Let X = Xi be a product of topological spaces, equipped with either the product or the i∈I box topology, A a topological space and f : A → X a function with coordinate functions given by f (a) = (fi (a))i∈I . (a) Prove that in this case, for both the product and the box topologies, if f is continuous then all fi are continuous. (Hint: For this direction same proof as in HW5, Q6(d), works.) (b) Find an example, where all fi are continuous, butQ f , for X equipped with the box topology, is not. (Hint: For example, consider Rω := R and the “diagonal” f : R → Rω , i∈N f (t) = (t, t, . . .). Can you construct a set, open in the box topology, with pre-image {0}?) Q Xi be the (5) Let {Xi }i∈I be a family of topological spaces with topologies TXi and X = i∈I product space. (a) Suppose that for i ∈ I, Bi is a topology basis for the topology on Xi . Prove that ( ) Y B= Ui : Ui ∈ Bi and Ui = Xi for i 6= i1 , . . . , in i∈I is a basis for the product topology on X, and B = Q Ui : Ui ∈ Bi a basis for the box i∈I topology. (b) Let {Xi }i∈I be a family of Q topological spaces and Ai ⊆ Xi subspaces. Show that the Ai coincides with the topology induced from X. product topology on A = i∈I Q (c) Prove that if for all i ∈ I, Xi is a Hausdorff space, then X = Xi , equipped with the i∈I product topology, is Hausdorff. Using a the box topology being finer than the product topology, deduce that the same holds for the box topology. Q (6) (a) Let I = [0, 1] and X = R, i.e. an element of X is a function f : I → R. i∈I (i) Prove that a sequence {fn }n ⊆ X of real functions converges to some f ∈ X in the product topology on X, if and only if it converges pointwise, i.e. for every x ∈ I, fn (x) → f (x) in the usual sense of convergence of sequences. (ii) Prove that if fn → f in the box topology, then fn → f uniformly? Construct an example when fn → f uniformly, but not in the box topology. (Hint: for the second task, think whether fn ≡ n1 is convergent in the box topology). Q ω (b) * Let X = R = R, the collection of all real sequences. Let A ⊆ X be the set of n∈N “eventually zero” sequences, i.e. A = {{xn } : ∃N.∀n ≥ N. xn = 0}, and BN = {{xn } : ∀n ≥ N. xn = 0}. Find the closure A and BN for X equipped with either the product topology or the box topology. 1