@1: show that the collection B = { [a,b): acb, a and b topology on R. basis for is Is the topology TCB) generated by B equivalent to the lower limit topology Justify your answer
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- Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X such thatX is T1. Please correct explanation.thanksDefine a collection T of subsets of Z+ as follows:W ∈ T if and only if n ∈ W implies that all positive divisors of n are also elements of W. Verify that T is a topology on Z+. In this topology find Cl({1}) and Cl({2}).
- Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝwhere ℝ_d denotes ℝ in the discrete topology. Compare this topology with the standard topology on ℝ^2.True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with the euclidean topology. The open intervals (0, 1) and (2, 3) are homeomorphic.3. a) We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in S. To see the non-compactness of S in another way, find an open cover of S that does not have a finite subcover! b) Is the Cantor set closed? Is it compact? Explain!
- 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.- Give an example of a closed subset of R that is not sequentially compact -Give an example of a metric on R^2Let X={a,b,c,d} Let T1={X, emptyset, {a}, {a,c}, {b}, {b,c}, {a,b},{a,b,c}, {b,c,d}} T2={X, emptyset, {b}, {b,d}, {c}, {c,d\}, {b,c},{b,c,d}, {a,b,c}} how that the T1 and T2 topologies are topologically equivalent (you can give the homeomorphism or show that they are by their bases)