Topology Problems

TOPOLOGY Question I. Show that defines a metric on IR2 • Question 2. Let (X, g) be a metric space. Show that for all x, y, z EX le(x, z) - g(y, z) I :S e(x, y). Question 3. Let A be a non-empty set of real numbers with infimum (greatest lower bound) b. Prove that there is a sequence, (xn LeN> in A whose limit is b. Question 4. Let p be a prime number. For each n E N* := N \ { 0}, let Vp ( n) be the exponent of p in the prime factorisation of n. This is the highest power of P which divides n. In other words, vp(n) = j if and only if (i) pi divides n, but (ii) pi+l docs not divide n. For each x E Q* := Q \ {O}, there are m, n EN* with Ix! := m. Define n with !xi := 􀂲 as above. Finally, define A : ,n, X ,n, -► ]Ro+ , (x,y) f---7 {Q ( ) "'7' ",! ",! p-Vp X-y Show that (Q, dp) is a metric space. if X = y if X # y. This is the p-adic metric on (Ql, which is important in number theory. TOPOLOGY Question 5 Let (X, e) be a metric space. Prove that g: Xx X -4 Rci, ( ) e(x,y) x,y -4 ( ) . 1 + g x ,y is also a metric on X Let (Y, a) be any metric space. Take functions f: X -4 Y and g: Y -4 X. Prove that (a) f is continuous with respect tog if and only if it is continuous with respect to (!; (b) g is continuous with respect to l! if and only if it is continuous with respect to (l. Question .J> Let (X, e) be a metric space. Taking IR with its Euclidean metric, <'-, and X x X with one of the canonical metrics on the Cartesian product. Prove that e: X x X -4 IRt, (x, y) i---+ e(x, y) is continuous. Given metric spaces, (X, e) and (Y, a), the function f: X -4 Y is uniformly continuous if and only if given any c > 0 there is a J > 0 with a(i(u), f (v)) < c whenever e(u, v) < J. Is e: X x X -> Rt uniformly continuous with respect to the Euclidean metric on JR and the product metric on X x X? Question} Let E be the Euclidean metric on Rn and let A = [aij] nxn be a positive definite symmetric real matrix. Define (x,y) t---+ L %(Xi -yi)(xj - Yj) i,j=l where x =(xi, ... ,x.,) and y =(Yi, ... ,Yn)- (a) Show that (Rn ,g) and (Rn ,<'-) arc isometric metric spaces. (b) Show that the function F: (Rn,€) -4 (Rn , g), X t---+ X is continuous. Question. (i) Show that the complement of any finite subset of a metric space is open. (ii) Show that every subset of a metric space is open if and only if each singleton subset is open. TOPOLOGY Question q Prove that the function max: JR x JR--+ JR, (x,y) 􀂩 -> max{x,y} is continuous, where JR x JR and JR have their Euclidean metrics. Question t. 0 Find a continuous bijection between topological spaces whose inverse is not continuous. Question i. I Let (X, T) be a topological space and take As::; X. Prove that there is a maximal open subset of X disjoint from A. This is the exterior of A, denoted cxt(A). Question f. 2. Let JR[t] denote the set of all real polynomials in the indeterminate t. Let S be a subset of JR[t]. Define V(S) := {x E JR I f(x) = 0 for every f ES}. Define F := {V(S) I S 􀀍 JR[t]}. Prove that there is a topology on JR, whose closed sets are precisely the elements of F and that this topology is not metrisable. This is the Zariski topology on JR and arises in algebraic geometry. Question\ 1 Let X be a set. Show that TOPOLOGY T := { A 􀀒 X I A = 0 or X \ A is finite } . is a topology on X. This is the finite complement topology on X. When is it metrisable? Question I- 'j- Let (X, 7), (Y,U) be topological spaces. Take X x Y with its product topology. Given b E Y, define Endow im(ib ), the image of ib, with the topology induced by the product topology on X x Y. Prove that is a homeomorphism. Question t· S Let JR and JR2 have their respective Euclidean topologies. Take s1 := { (x, y) E ]R2 I x2 + y2 =- 1} with the subspace topology induced from JR2 • (i) Define the relation~ on JR by a ~ b if and only if a - b E Z. Prove that ~ is an equivalence relation on R (ii) Let [a) be the ~-equivalence class containing a. Put lR/z := { [a] I a E JR.}. Endow IR/z with the quotient topology, that is, the topology induced by the natural projection r,: IR􀂵 R/z, a>--+ [a]. Prove that lR/z is homoeomorphic with S1 . Question l'- Take IR2 with its Euclidean topology. TOPOLOGY Prove that S1 = {(x, y) I x2 + y2 = l} c R2 is compact with respect to its subspace topology. Question 1· 1 Let JI be the closed unit interval, [O, l] := {t E JR IO$ t $ l}, endowed with its Euclidean topology. A path in the topological space (X, TI, from a EX to b EX is a continuous function 1:ll-+X with 1(0) = a and ,(1) - b. The subset A of the topological space (X, T) is path-connected if and only if for all a, b EA, there is a path joining a to b in A. (i) Prove that every path-connected set is connected. (ii) Determine whether the subset {(x,sin(􀃦)) IO< x $ l}U {(0,y) 1-1 $ x $ I} X of !R2 is (a) connected, (b) path-connected. Question f. g Let {(X).,U).) I>. EA } be a family of topological spaces. Let TIX). be the (Cartesian) product of the sets X). (>.EA). For each μ E A, let pr,., be the natural projection onto the μth factor pr,.,. : II X). -+ Xμ , (xAhEA >---+ x,,,. AEA The product topology on TIX). is the topology induced by {prμ Iμ EA}. (i) Prove that G 􀀔 TIX). is open in the product topology if and only if G= LJ Ga aEA for some indexing set A, with each Ga of the form Goe = II Ga). ACA where Goe). an open subset of the X). and Goe). = X). for all but finitely many >.s. (ii) Prove that for eachμ EA, the canonical projection, prμ : Il X). ➔ Xμ is an open mapping, that is, if G is an open subset of TIX). , then pr,.,.(G) is an open subset of X,,,. TOPOLOGY Question l- 'f Recall that (X, T) is a T4 topological space if and only if given disjoint closed subsets, K and L of X, there are disjoint open subsets, U and V, of X with K <; U and L <; V Prove that (X, T) is a T4 topological space if and only if given any closed subset F of X and any open subset, Hof X with F <; H, there is an open subset, G, of X such that FC GCGC H Question l!O Prove that every metric space is normal. Question ..2, f Let (X, e) and (Y, o-) be metric spaces. Prove that if f: X --t Y is uniformly continuous, then it maps Cauchy sequences to Cauchy sequences. Question I!. '2.. Let (X, e) and (Y, o-) be metric spaces. Prove that if X is compact, every continuous function f: X --t Y is uniformly continuous.