Which of the following is not a base for any topology on R? a. {(a, b]: a,b ER and a
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![Which of the following is not a base for any topology on R?
a. {(a, b]: a,b ER and a <b}
O b. {[a, b]: a,b ER and a <b}
O c. ((a, b]: a,b EQ and a <b}
O d. (R}
O e. {(a, 0): a EQ}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b91ba52-7254-4f89-a8fd-1284cdfbac07%2F39f32120-91c0-41e4-98b0-02c23e884973%2Fvftrnr_processed.jpeg&w=3840&q=75)
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- Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not ontoPlease answer the Exercise 1. Use the same proving technique used in showing ? = ℘(?) is a topology to show that ℳ = ℘(?) is a ?-algebra
- Define a cofinite topology T on an infinite set X. Answer each of the following for (X, T): (i) Verify that J is a topology on X;Let τs and τ be the standard topology and the countable complement topology on R, respectively.Determine whether (R, τs) is homoemoprhic to (R, τ ) or not.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.
- 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".Let Z be the set of all integers and let R be equipped with euclidean topology t prove that tr the topology induced on Z by t on R is the discrete topologylet (X,T) be a topological space and A,B nonempty subsets of X with A∩Fr(B)=∅If A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅
- 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.Show that the functions f, g : D^1 → D^1, f(x) = x^2 , g(x) = 1/2sin(x) are homotopic, where D^1 is the closed unit disc in E^1 and E^1 is R equipped with euclidean topology.I, Let ¥ ={a,b,c} and B={ {a,c} ,{b,.c} } c P(X). Show thatcannot be a base for any topology r on X . 2. Let (Vr) be a topological space. Where Y ={a,6 ,¢ ,d,e } andr={X .®,{c},{d}. {ed} .{d.e} .{e.d.e}, {b,c,a}, {a,b,c,d }}Show that f° ={ {c.d},{d,e},{a,b.c}} is a subbase for thetopology +r. 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".