Topology Questions

Question 1.
Let R x {0,1} be with the standard topology. Let X be the quotient space obtained from R x {0,1} by identifying (x,0) with (x,1) for every real number x with |x| > 1. Is X Hausdorff? Justify your answer rigorously.

Question 2.
Let R3 be with the standard topology. Let X be obtained by removing countably many lines from R³. Is X connected? Justify your answer rigorously.

Question 3.
Let C1 and C2 be two concentric circles in the plane and let X = C1 ∪ C2 . Let C1 be the inner circle and C2 be the outer circle. Let Ɓ1 be the collection of all single point sets formed by points in C2 . Let Ɓ 2 be the collection of sets of the form U ∪ V where U is an open arc on C1 and V is the radial projection of U on C2 with the midpoint (on C2 ) excluded. Let Ɓ 0 = Ɓ1∪ Ɓ 2 and let Ɓ be the collection of all_nite intersections of elements of Ɓ 0 .
(i)    Prove that Ɓ is a basis for a topology T on X.

For the following give rigorous proof/justification:
(ii) Is X connected?
(iii) Is X Hausdorff?
(iv) Is X compact?
(v) Is X separable?

Question 4.
Let X denote the diadic rationals (those rational numbers which have the form m/2ⁿ for some integers m,n). Let Y = Q \ X. Let I = R \ Q. Here Q and I are the sets of rational and irrational numbers on R, respectively. Note that each of the sets X, Y and I is dense in R with the standard topology. We define a finer topology T on R as follows: in addition to the standard open sets, we declare as open X, Y and all sets of the form {a} ∪ {w ∈ X ∪ Y: |w-a|< δ} and all possible finite intersections and arbitrary unions between those sets and those sets and the standard open sets. You don't have to prove that this is a topology on R. Prove that R with this topology T is connected.

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