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.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Topology Questions
    $40.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Topology Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats