Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If ACB, then AC B. indi (3) A is the smallest closed superset of A.
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- Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.Which vector spaces are isomorphic to R6? a M2,3 b P6 c C[0,6] d M6,1 e P5 f C[3,3] g {(x1,x2,x3,0,x5,x6,x7):xiisarealnumber}True or False: a)Every subset of a topological space is either open or closed.b)If X is a topological space with the discrete topology and if Xhas least two elements, then X is not connected.c) True or False: If X is a topological space, then there always is a metric on Xwhich gives rise to its topology.d) True or False: If X and Y are topological spaces and if f : X → Y is a constantmap (which means that there is a point y ∈ Y such that f(x) = y for all x ∈ X),then f is continuous.e) True or False: If X is a topological space, then X is both open and closed
- Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both infinite?Let (X,T) be a topological space Property C=P.C. A subset A of x has P.C If it's subset of the union of two disjoint nonempty open subsets of X then A is contained in only one of these open sets. Prove the following; If A and B have P.C and A̅̅∩B≠ø then A∪B has property c.(a) Supply a definition for bounded subsets of a metric space (X, d). (b) Show that if K is a compact subset of the metric space (X, d), then K is closed and bounded. (c) Show that Y ⊆ C[0, 1] from Exercise 8.2.9 (a) is closed and bounded but not compact.
- Do the following task.a) State formally what do we mean by being a metric space of X with metric d.b) State formally what do we mean by being a complete metric space of X with metric d.c) Let (X, d) and (X, d') be a metric spaces such that there is N ∈ N satisfying 1/Nd'(a, b) ≤ d(a, b) ≤ d'(a, b), (∀a, b∈ X). Show that if (X, d') is a complete metric space, then (X, d) is also a complete metric space.Show your working in each case: (a) Determine all the cluster points of the set M = {2 − 1 m : m∈N}. (b) Given the taxicab metric space Q2,τ, find the closed ball and radius r = 1. Sketch your solution. B(a,r) with centre a = (2,3) (c) True or false? The taxicab metric space Q2,τ, has infinitely many cluster points. (d) True or false? The taxicab metric space Q2,τ, is complete.What is Sierpinski Space? Also give example
- If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this theorem must be true:If (X,Ƭ) is not Hausdorff, then X is not a metric space. 1) Consider (ℝ,Ƭ) with the topology induced by the taxicab metric. Using the definition for Hausdorff, give an example of why (ℝ,Ƭ) is Hausdorff. 2) The finite complement topology on ℝ is not Hausdorff. Explain why ℝ with the finite complement topology is non-metrizable.Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.
