Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB, then A C B. (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.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 closedTopology For each of the following, if the statement about a topological space is always true, prove it; otherwise, give a counterexample
- 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.Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is constructed via Carath ́eodory’s theo- rem (Theorem 1.20). Corollary 1.24. Let (X, M , μ) be a measure space obtained via Theo- rem 1.20. Then (X, M , μ) is complete. Theorem 1.20 (Carath ́eodory’s theorem). Let M be as above. We have (1) M is a σ-algebra.(2) ForE∈M,defineμ(E):=ν(E). ThenμisameasureonM.What is Sierpinski Space? Also give example
- (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.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.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.